So to start, just to be clear, Lincoln didn't expel any Senators, for the reason that no president can expel Senators - it is a decision that can only be made by two-thirds of the Senate.

Now, the Senate link is a condensed version of information from Anne Butler and Wendy Wolff's United States Senate Election, Expulsion, and Censure Cases, 1793-1990, which is freely available online, so here is additional information.

The cases in particular involved Senators from five Southern states that by January 1861 had seceded from the Union: South Carolina, Mississippi, Florida, Alabama and Georgia. The Senators from these states (all Democrats) issued statements of withdrawal: Albert Brown of Mississippi on January 12, David Yulee and Stephen Mallory of Florida, Clement Clay and Benjamin Fitzpatrick of Alabama, and Jefferson Davis (yes, that Jefferson Davis) of Mississippi on January 21. They were followed in withdrawals by Alfred Iverson of Georgia on January 28, and John Slidell and Judah Benjamin of Louisiana on February 4. In addition to these formal withdrawals, there were informal ones: Robert Toombs of Georgia never officially notified the Senate, and James Hammond and James Chestnut of South Carolina told local newspapers.

On January 22, the Senate (including the Southern senators who had not yet withdrawn) met to debate just what happened. At issue may seem like a matter of semantics: did the senators withdraw or resign? However, the arguments behind the semantics were deadly serious - a withdrawal, as all the Southern senators in question claimed they were doing, implied that they were leaving the Senate because their states had seceded. Recognizing this implied that the state secessions were legitimate. Recognizing them as resigned would not recognize secession as legitimate, and would deem their seats as vacant, rather than nonexistent. The Senate debated, but could not come to an agreement, and tabled the discussion to March 14, after Lincoln's inauguration, and when the Senate gained a Republican majority in the new Thirty-Seventh Congress.

The new Senate turned down motions by Democratic members to recognize the withdrawals, or to note that the Senators in question "ceased to be members of the Senate", and instead decided to declare the seats of six Senators vacant: those of Brown, Davis, Mallory, Clay, Toombs, and Benjamin. Yulee, Fitzpatrick, Iverson, Hammond and Slidell weren't included because their terms had expired with the Thirty-Sixth Congress. Chesnut was dealt with in a case in July 1861 that we will turn to next.

This case came about because of the next round of state secessions. In March, Louis Wigfall of Texas notified the Senate that he understood that Texas had seceded, and that he was now a foreigner in the United States - but that he would remain in his office acting as Senator until he received official confirmation of secession. This ignited a debate in the Republican-majority Senate for expulsion of Wigfall, with Southern Democratic Senators echoing arguments made a month or two previous that it should be treated as a withdrawal due to secession, or at best to be treated as a vacant seat (as the earlier seats had been treated). The matter was conferred to the Senate Judiciary Committee, which did nothing.

By July 1861, the Civil War was in full swing, and the matter was revisited. By now, the matter taken up was not just Wigfall, but the other Senators of now-Confederate states who had withdrawn: William Sebastian and Charles Mitchel of Arkansas, Thomas Clingman and Thomas Bragg of North Carolina, Chestnut of South Carolina, A.O.P. Nicholson of Tennessee, John Hemphill of Texas, and James Mason and Robert Hunter of Virginia. I should mention that the other Senator from Tennessee - future Vice President and President Andrew Johnson - was a staunch Unionist and stayed in his Senate seat until 1862, when he resigned to serve as Lincoln's Military Governor of Tennessee.

Anyway, on July 10, Republican Daniel Clark of New Hampshire introduced a motion to expel the ten Senators in question, in effect for conspiracy against the United States by failing to appear in the Senate and perform their duties. Democrats James Bayard of Delaware and Milton Latham of California argued against expulsion, stating that it implied personal misconduct or moral turpitude, and that the Senators in question were acting on instructions of their state governments (remember this was when Senators were elected by state legislatures). Ultimately, the mood was militant, and the Senate voted the following day on expulsion, 32 to 10.

So: six Senators had their seats declared vacant in March 1861, five Senators had terms that expired in March, and another ten were expelled in July, with the lone Johnson keeping his seat. None of this was specifically because of Lincoln's electoral victory, but because of state declarations of secession - the senators in question felt that this terminated their states' participation in the Union, and all withdrew. In the case of the ten expelled, the expulsions were pro forma, and largely meant as a gesture against the Senators in question as the war was already well under way.

Of the withdrawn Southern senators, here are major roles they played in the Confederate government: Davis obviously became Confederate president, Toombs Secretary of State, Benjamin Attorney General, then Secretary of War, then Secretary of State, Hemphill delegate to the Provisional Confederate Congress, Wigfall also a delegate, then a Confederate Brigade Commander and Senator, Mitchel a Confederate Senator, Mason a Confederate diplomat involved in the Trent affair, Hunter Secretary of State, then Senator and President Pro Tempore of the Confederate Senate, Clingman a Brigadier General, Bragg Attorney General, Chestnut a Provisional Congress Deputy and later a Confederate officer attaining the rank of Brigadier General, Clay a Confederate Senator and an informal head of Confederate sympathizers and agents in the Great Lakes area, Brown a Confederate Senator, and Mallory Secretary of the Navy. Only Sebastian and Nicholson did not play major roles in the Confederate government.

I want to add a little bit to the great answer by /u/Kochevnik81 to your original question.

One things worth noting are that various fights over seating members of both Houses of Congress weren't just a Civil War era phenomenon; in fact, one of the first truly nasty bits of electoral skullduggery was in 1793 when Federalists expelled Albert Gallatin - later Secretary of the Treasury under both Jefferson and Madison - over a pretty dubious claim about his eligibility because of his naturalized citizenship. (He was a Swiss immigrant, but that was far less important than being a Jeffersonian Republican and one of the few people in the country good enough at public finance to go toe to toe with Hamilton.) James Monroe, at the time the Jeffersonian Republican leader in the Senate, returned the favor tit for tat by blocking the seating of Federalist Kensey Johns of Delaware, and Gallatin was elected to the House two years later, seated, and promptly began investigating the Administration's finances.

But this set the pattern for what Congress could do to itself as by law it is the arbiter of its own membership, and while the Early Republic was as nastily partisan as any era in history, it also served to both parties as a warning that anything one party could do, the other party would as well - which actually did reduce a little bit of the electoral bickering on the Congressional level, albeit with every few years a disputed election or two that got fought out in the caucuses and on the floor. As I've written before, one of the more memorable of these was when 4 of the 5 New Jersey House seats had dueling Whig and Democratic slates certified by relevant state authorities, which meant party control of the House was in question, it couldn't organize to elect a Speaker, and rather than letting the non elected clerk run things, both parties agreed on the somewhat non-partisan John Quincy Adams to chair proceedings until they could sort out the election and membership.

After the Civil War, while readmission of individual states and their Members of Congress are well beyond your question since it gets deeply into the details of Reconstruction, one worth noting was that when Radical Republicans needed a 2/3rds majority to pass the 14th amendment in 1866, an excuse was found to remove a Democrat from New Jersey, Senator John Stockton - who'd already been sitting as a Senator for a year - over the rather thin veneer that since his election was by plurality rather than majority it was therefore invalid. In one of the most bizarre procedural moves in Congressional history, he was expelled, his voting record over that year entirely expunged (even more strangely, during all this he'd actually been allowed to vote on parts of the investigation into his status), and as Republicans now controlled the New Jersey legislature, one was elected by them for remainder of his term. Stockton got a bit of revenge when Democrats regained control of the state a couple years later and he was reelected.

Your followup question, however, is one that I've discussed a few times before in the context of Congress and the Electoral College. The Election of 1800 set the tone for the disasters to come and nearly blew up the nascent republic, 1872 was the only time in history Congress outright refused to accept electoral returns, and 1876 had Republicans commit massive recount fraud to place Hayes in office, and had Tilden not backed down/bumbled away his claim there was a decent possibility of competing inaugurations, along with a lesser one of McClellan marching on Washington on his behalf to fight Republican militias and perhaps federal troops

I think what you are really after when you reference January 6th, though, is a discussion of what Congress enacted after the 1876 debacle, the Electoral Count Act of 1887 (note the 11 years it took to pass!) and how it defines Congressional responsibilities when the Electoral College results are actually tabulated. That's a bit much for a followup question, but if you'd like to ask it as another top level question in a couple of days I might be able to give a more thorough answer on it.