The victory moved Exeter into fifth place in the Premiership table, level on points with sixth-placed Quins.
The Chiefs go to Leicester on the final day of the season while Harlequins host seventh-placed Bristol.
Should Sale lose to Saracens or Bath fail to get a bonus point at home to Northampton – who have already secured a home semi-final – Exeter, Harlequins or Bristol could make the top four.
For Exeter’s new-look side it means an exciting end to a season that has seen a mix of high and lows.
“This is all I could ask for,” said Baxter.
“Even two or three weeks ago I was at home saying to my wife ‘if we can just get into that last couple of games with something worth fighting for is going to be incredible’.
“So to go into the last week with something worth fighting for is going to be amazing.
“It makes the whole season worthwhile for us, and I just hope we get plenty of support up there and I hope we can get our feet on the ground quickly enough to make sure we get up there and at least do ourselves proud.”
