Substitute lock Aly Muldowney's last-gasp try, which was awarded on the judgment of television match official Tony Spreadbury, and converted by Muldowney's fellow replacement Gareth Steenson, saw Exeter edge out Gloucester 28-27 at Kingsholm.
It kept them close on the heels of Northampton and Leicester, while Exeter also strengthened their Heineken Cup qualification hopes and they still have an Amlin Challenge Cup quarter-final to relish in a fortnight's time against Stade Francais.
"Our attitude was first class today," said head coach Baxter, following a gripping triumph.
"The only way you win in the Premiership is by going after games, and our endeavour to play was fantastic. In the end, the scoreline went our way.
"We have never made any secret to our approach in that we will take every game is it comes, and if the rewards come then we will enjoy them.
"We don't spend a great deal of time looking at league tables - we did it for a short time last season, and then lost two games in a row - or look at what other teams are doing."
Exeter looked to have blown it when Gloucester staged a stunning fightback from 18-6 adrift approaching the hour mark as they scored three converted tries during a devastating seven-minute burst.
Wing James Simpson-Daniel, fly-half Freddie Burns, courtesy of a brilliant solo try when he caught his own kick ahead one-handed, and replacement scrum-half Nick Runciman all touched down, while goalkicker Burns finished with 17 points.
Exeter seemed destined to head back down the M5 with only a losing bonus point, secured by Steenson's late penalty, and they would have been kicking themselves after prop Craig Mitchell and wing Matt Jess claimed first-half tries.
Fly-half Ignacio Mieres added two penalties and a conversion, but just when it appeared as though Exeter were finished they dug deep to claim a first win in their history at Kingsholm.
"We are in a nice place as a club," Baxter added. "The players work very hard and we are playing some great rugby.
"If we get to a Premiership semi-final, then fantastic, but we have got some tough games coming up.
"The important thing today was to take the game to Gloucester, and winning here backs up some of the things we have been saying to the players.
"When we came here last season, Gloucester had scored three or four tries before we had even got off the coach."
Gloucester head coach Bryan Redpath refused to point to the TMO's debut in a non-televised Premiership game as the reason for his team's late demise.
A TMO was in place at Kingsholm as part of a four-fixture experiment that could be rolled out across every Premiership match next season.
Only games televised live currently have a TMO, but Premiership clubs might expand that for all league fixtures next term at a cost of around £350,000, with Gloucester's clash against Exeter launching the trial period.
Redpath smiled ruefully when asked about the injury-time drama, but he said: "It would be sour grapes of me to use the TMO as an excuse.
"Our first-half performance was not good enough. We weren't at the races in those first 40 minutes, physically or mentally.
"We gave away three penalties at 27-18 up, and that was massively disappointing. Realistically, that is it for us now in terms of the top four.
"We have got four games left to get in the top six and secure qualification for the Heineken Cup.
"We've lost today on a last-minute decision, but that was not our biggest problem. We have not been consistent or clinical enough all season to finish off tight games.
"We have had too many games this season that we have not been able to see out."
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