Glasgow broke their duck in this season's Champions Cup with a 28-21 bonus-point win that dissolved Exeter's chances of finishing as one of the best three pool runners-up.
A hard-fought first half dominated by Exeter left the scores tied at 7-7, with an early try from Stuart Hogg matched by a late one from Sam Simmonds.
Glasgow were awarded a penalty try on 53 minutes and with Chiefs scrum half Nic White in the sin bin the home side ran in tries from Tommy Seymour and Matt Fagerson.
Chiefs hit back with tries from Don Armond and Ian Whitten, but Glasgow held them in the closing stages.
Exeter kicked off on a pitch cleared of snow just minutes before however it was Glasgow's man of the moment Hogg, returning to action after a long injury absence, who grabbed the limelight touching down under the bar after a break by Finn Russell who converted.
However it was the best part of 30 minutes before the visitors' 22 was troubled again. Exeter dominated possession and territory but focused defence kept the Glasgow line intact.
In the 20th minute Exeter seemed to have scored. Glasgow lost hold of the ball at the back of their scrum. It was moved left and Whitten appeared to have driven through a tackle for the score.
However, referee Romaine Poite called in the TMO and ruled the ball had not been grounded.
Hogg then chipped into space, hacked on and only Gareth Steenson somehow getting a hand to gather the ball under his bar prevented a Glasgow score.
As the half ended further Exeter pressure ended with a mini maul heaving number eight Simmonds over the line for Steenson to convert and the sides go in level.
Glasgow made a bright opening after the pause and got back in the lead with a penalty try on 53 minutes. As Glasgow pressed from a close in scrum referee Poite had judged Exeter scrum-half Nick White to have deliberately knocked on and he went to the sin bin.
As the fourth quarter started Glasgow scored two similar tries in two minutes. The Chiefs' left-side defence was shredded on each occasion with the first try falling to Seymour and the second to Fagerson with Russell converting both.
Chiefs soon hit back with a pick-and-go score from Armand followed by one run in by Whitten, with both converted by Steenson.
