Moyne broke Bristol hearts with a try in injury time
Harlequins scored an injury-time try to stop the Premiership's bottom side Bristol getting off to a winning start under new head coach Paul Hull.
Bristol led at half-time thanks to a converted Neil Brew try although Gary Botha pulled the deficit back.
David Strettle put the visitors ahead with a try before Bristol were awarded a penalty try after David Lemi was barged as he attempted to touch down.
But Ugo Monye scored at the death to win it, Chris Malone converting.
It denied Bristol, who parted company with coach Richard Hill this week, only their second league victory of the season despite a battling display at the Memorial Stadium.
A losing bonus point proved scant reward from a game that leaves Bristol 12 points behind 11th-placed Worcester with just seven Premiership fixtures left.
Bristol head coach Paul Hull:
"The last couple of weeks have been difficult circumstances.
"But the players have responded very well. I wanted to up the ante today, and we got ourselves in a position to win.
"We are very disappointed, though, we didn't finish things off.
"Worcester and Newcastle have turned the corner in the Premiership, which makes it difficult for us, but we just look at our performance and focus on what we can do.
"We want to finish the season with a couple of wins under our belts. We can still try to grind out some wins."
Harlequins director of rugby Dean Richards:
"I thought we showed a lot of endeavour in the first half and tried to play, but we didn't quite get the rub of the green. It became a very messy game.
"I will be looking very closely at the referee's decisions and consistency across the board.
"The disappointing thing was the way he didn't take charge of the scrums."
Bristol: T. Arscott, Robinson, N. Brew, Eves, Lemi, Jarvis, Perry, Irish, Blaney, Crompton, Attwood, Budgett, R. Pennycook, El Abd, Ward-Smith, Thompson.
Replacements: Barnes for Jarvis (4), Winters for Attwood (47), Thompson for El Abd (67), To'oala for Thompson (77). Not Used: Regan, Thomas, N. Brew.
Replacements: Monye for T. Williams (65), Poluleuligaga for Gomarsall (73), McMillan for Ross (70), J. Evans for Kohn (61), Lambert for Robshaw (67). Not Used: Croall, Luveniyali.
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