I've been watching Rome recently and I asked myself this question and wasn't really satisfied with what I could find on Wikipedia. Did it have any really purpose after Octavian took over? When did it end?
It did indeed continue to exist. I'm not an expert on the subject though, so I'll just redirect you to this similar question, which was answered by /u/shlin28, who is an expert on the subject: https://www.reddit.com/r/AskHistorians/comments/4i9ca4/what_role_did_the_byzantine_senate_play_in_the/
The Roman senate never ceased to be after the Empire was founded. In fact Augustus (the first emperor) was politically smart enough that he still involved the senate in many of his decisions. The senate was still 300 of the wealthiest men in the empire so excluding them from power would've been a terrible decision.
In fact Augustus didnt even call himself emperor. He preferred the title Princeps (first citizen). Over time as the Empire became more "normal" (instead of people wanting to go back to a republic) some emperors would exclude or punish the senate and these emperors more often than not would be assassinated. Domitian is probably the best example as he actively tried to piss the senate off. As a result of this the senate had him assassinated and damned his memory. It wasn't until the 20th century that we were able to find out what kind of emperor Domitian actually was because the senate (not as a body, but individual senators with grievances) basically tried to scrub him from the history books as a bloodthirsty tyrant.
So depending on the emperor that was reigning the senate still held a lot of power. Trajan, who is considered to be one the top 3 emperors in Roman history actually had a policy of deferring decisions to senators or governors because he knew that they knew their circumstances and people the best. It wasnt until the capital of the empire was moved to Milan that the roman senate truly lost all power in decision making in the Empire.