Scott Wight missed simple late penalty as Glasgow fell to a disappointing home defeat to Treviso at Firhill.
Duncan Weir scored a fantastic first-half try and converted it but three penalties from Kris Burton had the visitors in front at the interval.
Weir added another penalty after the break and White kicked the home side in front again after Burton had replied.
Burton won the game with an untidy drop-goal on 72 minutes, with White skewing his late effort wide.
Warriors had been aiming to maintain the momentum they had gained after last week's impressive away win over Leinster, yet they put in a very flat performance, riddled with errors.
And the Italian side turned in a dogged display to earn their first ever away victory in the RaboDirect PRO12.
It was the hosts who were first to get off the mark, a flowing move creating sufficient space for Weir to squeeze over in the corner for a self-converted score.
Treviso stand-off Burton soon had his side on the scoreboard, however, atoning for an early penalty miss with a successful attempt on 20 minutes.
Midway through the first half, Glasgow were reduced to 14 men when hooker Pat MacArthur was ordered to the sin-bin for excessive use of the boot at a ruck.
Burton slotted over the resulting penalty and added another on the stroke of half-time.
Weir and Burton traded further penalties to ensure the Italians maintained their two-point advantage heading into the final quarter, despite having played a period with reduced numbers, with captain Antonio Pavanello serving time in the bin for repeated offences at the breakdown.
The home side spurned the chance to regain the lead after a period of sustained pressure when Weir sent a penalty attempt horribly wide.
Weir's replacement Wight secured the hosts a slender advantage with a confident penalty heading into the final throes.
But that advantage was short-lived, the boot of Burton once again proving to be accurate as he wobbled over a drop goal.
Glasgow pressed frantically in the closing stages, with Wight's drop goal attempt wide of the mark.
And it was the former Melrose man who had the last opportunity to snatch the win for his side, but from only 15 metres out his effort on goal sailed agonisingly wide of the left-hand upright.
Glasgow head coach Sean Lineen: "The effort is there but it's just the top three inches. There were some not good decisions out there.
"The pressure with the way Treviso play, or don't play - we didn't deal with it well.
"We missed five kicks at goal, so that's part of it, but there were just some bad decisions after going 7-0 up and that takes the momentum out of the game.
"The guys are trying. It's just rugby smarts [that are missing]. We have to learn from it and that is a harsh lesson."
Glasgow: Hogg, Aramburu, Horne, Nathan, Shaw, Weir, Pyrgos, Grant, MacArthur, Kalman, Ryder, Campbell, Harley, Fusaro, Wilson. Replacements: Seymour for Shaw (57), White for Weir (65), Gregor for Pyrgos (50), Cusack for Kalman (50), Eddie for Campbell (50), Beattie for Wilson (57). Not Used: Gillies, Reid.
Benetton Treviso: de Waal, Nitoglia, Galon, Morisi, Williams, Burton, Chillon, Filippucci, Minto, Vosawai, Vermaak, A. Pavanello, Di Santo, Vidal, Rizzo. Replacements: Picone for Williams (60), Padro for Chillon (54), Iannone for Filippucci (62), Bernabo for Vosawai (52), Allori for Di Santo (41), Muccignat for Rizzo (52). Not Used: Ceccato, Di Bernardo.
Ref: Alan Falzone (Italy).