They knew that with results elsewhere starting to go their way, a bonus point win could lift them into the top section of the league, so they set about hunting it straight away, turning down two long-range kickable penalties to go for the corner and tries.
It took a couple of goes but in the end it worked with Jonny Gray, the captain, crashing over from short range for the opening try just past the ten minute mark. With Finn Russell converting, that should have given them the cushion to push on.
Never, however, underestimate the Scots capacity to self destruct, and two sloppy penalties soon after the restart gave Jayden Hayward, the Treviso full back, a chance to pull back the deficit, which he took.
Glasgow did have another chance but Simone Favaro, the flanker, lost control of the ball reaching out to score against his former team-mates, though scrum half Henry Pyrgos quickly made amends, first winning a penalty and then tapping and racing away on his own. Though he was caught, Peter Horne was there to pick up and score.
The comfortable lead was soon wiped out, though. Treviso had always been willing to run with the ball and a quick throw in caught the Scots dozing as they shifted the ball across field where Sam Christie found the space to wriggle through and find Angelo Esposito, the wing, in support to run clear and score.
Bad was soon worse as Treviso took the lead when Hayward edged his side ahead with his third penalty just before the break.
It did not last long, though. The second half opened exactly as Glasgow must have hoped with their players throwing themselves into a long, sustained attack before Stuart Hogg danced his way through a crowd of tacklers and offloaded for Favaro to at last claim the try.
Treviso defended gamely as Glasgow went to their bag of tricks to try to unlock the defence without being able to find the secret key.
In the end, it all came down to another piece of Hogg magic as the full back s went outside the first line of defence, then had the pace and control to keep himself half an inch inside the touchline as he raced clear to score.
Job done, Glasgow started to flood the field with their replacements, though it did not disrupt them enough to lift the pressure and Grayson Hart, the replacement scrum half, did get a fifth try while James Malcolm, the replacement hooker got the sixth.
SCORERS: TREVISO: Try: Eposito (36). Con: Hayward. Pens: Hayward (14, 17, 39)
Glasgow Warriors: Tries: Gray (11), Horne (29), Favaro (42), Hogg (59), Hart (67), Malcolm (73). Cons: Russell 4.
Treviso: J Hayward; L Nitoglia (L McLean, 54), T Iannone, S Christie, A Esposito; J Ambrosini, E Gori (A Lucchese, 74); Alberti De Marchi M Zanusso, 48), L Bigi (D Gizzon, 61), F Filippetto (R Harden, 42), F Paulo, M Fuser (D Budd, 66), M Lazzaroni, A Zanni c (A Steyn, 54), R Barbieri (Andrea De Marchi, 54), .
Glasgow Warriors: S Hogg (G Bryce, 71); T Naiyaravoro (L Jones, 77), M Bennett, P Horne, S Lamont; F Russell, H Pyrgos (G Hart, 60); G Reid (J Yanuyanutawa), F Brown (J Malcolm,70), S Puafisi (D Rae, 66), T Swinson (L Nakarawa, 60), J Gray (Captain), J Strauss, S Favaro (T Holmes, 58), A Ashe.
Referee: B Whitehouse (Wales).