Gregor Townsend's side had not played since Boxing Day and were caught cold as Exeter bagged a try and a converted penalty inside the opening 10 minutes. The visitors led 12-10 by half time thanks to tries from Sean Maitland and Sean Lamont but Glasgow's hopes of staying in Europe hang on gaining a bonus-point victory away to pool winners Toulon on Saturday while hoping Exeter can give them some help when they face second-placed Cardiff Blues.
It was the Glasgow players' last chance to impress before Scott Johnson announces his Scotland squad for the Six Nations on Wednesday but a lack of action following the postponement of the second leg of their 1872 Cup double-header with Edinburgh appeared to take its toll.
Head coach Gregor Townsend said: "I think we did well in attack in the second quarter of the game and we also defended well in that quarter. But in the second half no-one really got into their game and it wasn't a very good spectacle. Maybe there was a rustiness. We made a couple of breaks and we didn't connect - we turned over the ball in their 22 three times. But we've had a game to get the cobwebs off and when we play Toulon we have nothing to lose."
Home fly-half Henry Slade opened the scoring with a third-minute penalty and then converted after winger Fetu'u Vainikolo scored between the posts under the noses of the Warriors forwards.
However, only a minute later a knock-on by Tom Johnson dribbled forward and Lamont picked up. Sweeping from sided to side, the ball landed with prop Ed Kalman and he then passed to Maitland, off his wing and on the other side, and the Lions player touched down in the corner for an instant reply.
The momentum was with Glasgow for the rest of the half and, after Ruaridh Jackson and Stuart Hogg bounced off some sturdy home tackles and their own front five attempted a few ill-advised offloads on the halfway line, the playmakers found another gap.
A skittled ball landed at the feet of Jackson, who threw a 20-metre pass to Maitland. Lamont came off his own wing and stretched every sinew to collect the ball and throw his body towards the line and score. Hogg converted to give Glasgow a 12-10 half-time advantage.
Hogg slotted a penalty early in the second half but little else of note happened until the 78th minute when the Chiefs earned a penalty due to the Warriors popping up in the scrum.
Ceri Sweeney, on for Slade, heaved the ball into the corner and the Chiefs caught and drove. Warriors, powered by substitute James Eaddie, pushed them into touch and won a line-out. There were seconds left and, with the ball at their own mercy, all they had to do was kick it out.
However, Niko Matawalu had other ideas. Heaving it into the early night sky, the unpredictable Fijian kept the ball in play and Chiefs were gifted a final chance. They inched towards the line, but Fusaro snatched up the ball and the visiting support let out a sigh of relief.
Glasgow: S Hogg; S Maitland, B McGuigan, A Dunbar; S Lamont; R Jackson (D Weir 61), H Pyrgos (N Matawalu 61); R Grant (J Yanuyanutawa 61), D Hall (P MacArthur 49), E Kalman (J Welsh 49), T Swinson (L Nakarawa 69), T Ryder, R Harley (J Eadie 61), C Fusaro (capt), J Strauss (R Wilson 14).
Exeter: L Arscott (capt); F Vainikolo, S Naqelevuki (I Whitten 51), S Hill, M Jess; H Slade (C Sweeney 65) , D Lewis (W Chudley 57); C Rimmer (B Moon 51), L Cowan-Dickie (G Bateman 68), A Brown (L Fairbrother 61), D Welch, J Hanks (R Graham 21), T Johnson (D Armand 57), J Scaysbrook, K Horstmann.
Referee: G Clancy (IRFU)