Baxter is hopeful his side can push on despite playing in front of a disappointing crowd of just 5,405 at Sandy Park.
"We have got a nice little run going and winning is very nice and it made everything pleasurable," said Baxter.
"It means that the boys can go to the bar and chat with the supporters and when they come in for training on Monday they will know that they have achieved something special and prepare for Saturday's visit to Bath.
"We talked at length about this season that we need to get better, that is the challenge for us, and it is not about trying to fight our way through a second season in the Premiership.
"It is about attacking something and everything we do - whether it be in training or our mental focus - needs to be a little bit better and that will hold us in good stead.
"For a lot of this game it was tough and I found it particularly hard to watch. I don't think we hit the game with the absolute focus or sharpness that we needed and we paid the price for that for quite a while.
"But to get the guys in at half-time and reset ourselves as well as we did know that we had to work extremely hard to get anything out of the game.
"We had to drag ourselves off the floor, we had to keep coming at Newcastle, masses of multi-phase play had to go on and we won the game because of that massive work rate and endeavour."
Baxter added: "Newcastle had 15 very committed men, they certainly weren't going to back down.
"I am obviously disappointed that we conceded a late try but we were chasing the bonus point then. We'd reset our goals to all-out attack and if you make a defensive slip you have to take that on the chin.
"But [after that try] we attacked the kick off, when we could easily have kicked the ball off the pitch for full time, and we were prepared to work for three of four minutes to force a try to come away with a bonus point."
The defeat was Newcastle's ninth in a row in the Premiership - their poorest run in the history of the tournament - and head coach Alan Tait accepted that you can have all the possession in the world but it is scores that count.
He said: "You work at a problem all week to sort things in defence but then we let them score from the very first play - almost as we did with Bath last week.
"But after that we played the best 38 minutes I have seen our lads play for a long time and get ourselves back in the game. In the second half I thought that we would kick on but Exeter have power runners.
"One individual like their man of the match [Sireli Naqelevuki] can just break down defences and get over the try line can give you a bit of breathing space. There were key times when we could have got back into the game but we didn't get it right.
"That is where we will have to sit down in the coming week and sort these things."
The Chiefs were quickly on the scoreboard with wing Josh Tatupu touching down after less than 90 seconds, with their Argentinian fly-half Ignacio Mieres adding the conversion and two penalties.
Newcastle replied with a try in the corner from England Under-20 player Tom Catterick, with fly-half Jimmy Gopperth adding a penalty on the stroke of half-time as the Falcons trailed 13-8 at the break.
Centre Naqelevuki barged through under the posts for the Chiefs' second try and five minutes from time a touchdown by full-back Phil Dollman under the posts was improved by Mieres when he kicked the conversion.
Falcons centre James Fitzpatrick scored a late consolation try, which fly-half Gopperth converted.
But with time up on the clock the Chiefs kept the ball alive and inside centre Jason Shoemark squeezed over in the corner for the final score of the game.
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