The chiefs completed the double over the east Midlands side with a 19-11 victory at Sandy Park moving them up to fifth place in their second season in the top flight and ending Leicester's run of five league wins.
Baxter said: "You have to get the record straight because we played them at Welford Road during the World Cup when they were severely depleted and we have been fortunate enough to play them during the Six Nations.
"So as great as it is to do the double over anybody, there are some factors that are involved and we have to be realistic about that.
"The truth is I enjoy and celebrate, and I tell the players to enjoy and celebrate every Premiership victory because they are still very important.
"It doesn't matter who you play, this is just a bit sweeter because it is Leicester and it is sweeter because you respect them so much.
"The players often talk about how you should attack teams who you respect and we say that you show them respect by going at them flat out.
"I think we did that in this game. Unfortunately we didn't just have passion, workrate and a great desire to win the game, we also put together great rugby, with enough composure to come out on top.
"Often when we have lost games, especially at home, to other Premiership sides it is because when the pressure is cranked up we have lost composure and given away a soft penalty or a soft try but we didn't do that today.
"I thought defensively we were very good, we created pressure with our defence which really told for us by the end of the game - and it's something that we are improving all the time.
"We are in a nice place at the moment, we don't have to worry about the bottom of the league. We shouldn't be talking about the fear of losing games, we should be talking about a desire to win things.
"That's what we are talking about now, it may be a little bit early in our Premiership lives to be doing that, but in this season in particular it is not. We have to try to push through and be ambitious and see what the season brings us."
Exeter will be without prop Brett Sturgess for some weeks after he had to leave the field towards the end of the game having broken his right arm after throwing himself into a tackle.
Leicester had a slender 8-6 interval lead after Samoan wing Alesana Tuilagi powered through for the only try of the half, with fly-half Toby Flood adding a penalty to two from his opposite number Ignacio Mieres.
After a string of Tigers penalties to the corner Flood finally slotted his second penalty, only for number eight Richard Baxter to race through to touch down in the corner for his fourth try of the season for the Chiefs.
Mieres' touchline conversion nudged the Chiefs in front for the first time to set up the prospect of an exciting final quarter. The Argentinian half-back landed two more penalties to seal the victory.
Leicester's director of rugby Richard Cockerill said: "We were okay at half-time, then we kicked a penalty - but then there was the charged down try and the whole thing turned against us.
"We didn't control the ball, we forced the game, played into their hands and we should have been better than that. But we weren't and they worked hard, you make mistakes and they punish you.
"That is exactly what they did and in the end they deserved to win. For whatever reason we started to chase the game with 20 to go and made mistake after mistake and put ourselves under pressure.
"We have only ourselves to blame and we'll deal with it on Monday because I am not accepting that performance, it's not good enough.
"We are not having that because we are better than that and we had opportunities to win the game."
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