Both teams scored a try apiece, but it was the kicking of Exeter's Ignacio Mieres that ultimately proved to be the difference maker.
The Tigers opened the scoring eight minutes in. A quick ball allowed Toby Flood to float a long pass out to Geordan Murphy, who moved it on to Alesana Tuilagi, allowing the Samoan to bash his way to the line for the try.
Flood was unable to convert from the touchline, but it gave an early signal of Leicester's intent.
The two sides traded penalties before Exeter had a try disallowed when it was ruled that winger Matt Jess was in touch following Manu Tuilagi's tackle.
Mieres missed one penalty then kicked another to bring the score at half-time to 6-8 to Leicester.
Flood also missed and then scored a penalty at the start of the second half to extend the Tigers' lead, but Exeter soon took the lead with a try of their own.
As Leicester tried to clear their lines, a kick was charged down and the Chiefs' Baxter won the race to loose ball to level the sides at 11-11, Mieres clotting the conversion to give the home side a two-point lead.
Two more penalties from the boot of the Argentinian kept the scoreboard ticking over for the Chiefs, while his opposite number didn't have the same luck, with Flood missing another penalty with less than ten minutes to go.
It would prove to be Leicester's last chance to get on the scoresheet as Exeter ran out 19-11 winners.