I could not say if he was the "first" candidate to do so, but Lyndon Johnson, in his 1948 race for Texas's vacant U.S. Senate seat against retired Governor Coke Stevenson, relied on a significant fundraising disparity to close the gap between him, and the massively popular Texas statesman.

Robert Caro's Years of Lyndon Johnson -- and especially book 2, Means of Ascent -- describes Johnson's political empire as a little-discussed network of major corporate donors, forming a veritable who's-who of major Texas industrialists: Sid Richardson (oil magnate and philanthropist [he was later a major donor to Rice University]) , Herman Brown and George R. Brown of Brown & Root (which would be acquired by Halliburton on the latter's death), and Alan J. Wirtz (businessman/politician). Money made its way to Johnson's campaigns through the ordinary means of above-board contributions up to the then-existing limit of $5,000.00; then, it reached him in cash-stuffed envelopes, hand-collected and hand-delivered by aides.

All of this meant that when Johnson declared against Stevenson, the latter's ordinary time-tested means of campaigning -- driving town to town, sitting and speaking with voters in town squares -- was quickly overmatched. By circumventing campaign finance caps, Johnson was able to blanket the Texas airwaves in a manner unprecedented in early-modern politics -- helped, of course, by the fact that he owned, through is wife, several major local radio stations -- and even hired two helicopters to take him on a jet-age version of a whistle-stop tour of the state.

Even that wasn't enough to win the race, which dissolved into allegations of vote-rigging and fraud, and ultimately had to be resolved in court when Johnson claimed to have "won" by under 100 votes. Caro makes a persuasive case, based on an interview with the only south-Texas party operative still alive at the time he wrote the book, that the decisive votes were in fact invented. It's a good and depressing read.

Note: the $5,000.00 limit Johnson worked under (and circumvented) was a product of the 1939 Hatch Act. The Federal Election Campaign Act of 1972 would lower the individual cap to $1,000.00 per candidate, with a per-cycle individual cap of $25,000.00. The per-candidate and aggregate caps were both found constitutional in Buckley v. Valeo, 424 U.S. 1 (1976). The aggregate cap was invalidated in the recent case of McCutcheon v. F.E.C., 572 U.S. ___ (2014), and the $1,000.00/candidate cap was increased by the "McCain/Feingold" Bipartisan Campaign Reform Act.

Money likely had a good deal to do with William McKinley's 1896 win over William Jennings Bryan. Bryan's threats of "Free Silver" (I believe the topic of his famous "Cross of Gold" speech) spooked wealthy East Coast industrialists and allowed McKinley's campaign manager, Mark Hanna, to amass the largest campaign chest in history, pegged at $3.5million.