An early penalty from Olly Barkley put Bath in the lead, and when new Scotland wing Simon Danielli went over for a try after 24 minutes, it looked as if Michael Foley's side might get the better of the visitors.
But as they have shown down the years, Leicester are made of stern stuff, and answered back two inutes later with a try from Ben Kay, the England lock who this week had been reported to be about to sign a lucrative three-year deal to keep him at Welford Road.
Ireland wing Geordan Murphy converted that score, and they took the lead when Sam Vesty notched a penalty midway through the second half.
That lead was extended when Murphy went over for a try with ten minutes to go, Bath still claiming a bonus point as the Leicester bandwagon gathers pace again at a crucial stage in the season.
For Bath:Try: DanielliPen: Barkley
For Leicester Tigers:Tries: Kay, MurphyCon: MurphyPen: Vesty