Substitute Matthew Whichelow secured the Hornets all three points with a 12-yard effort five minutes from time to stretch their unbeaten run to seven games.
The 19-year-old's winner came after an interrupted afternoon which saw power cuts threaten to put paid to the npower Championship clash.
Let there be light: Players keep warm in darkness as the fans watch on in the blackout
Martyn Woolford had put the Iron ahead in the 60th after the first power failure, making amends for Michael O'Connor's missed penalty moments earlier.
Watford, though, did not buckle and equalised through Danny Graham before - following another two power cuts - securing victory through Whichelow.
Scunthorpe made two changes heading into the clash, bringing in Garry Thompson and Josh Wright in place of Michael Collins and Sam Togwell.
The Hornets, meanwhile, were without captain John Eustace through a calfinjury sustained in Saturday's 3-0 drubbing of Portsmouth.
Ross Jenkins replaced the skipper in the starting XI, while Troy Deeney's recovery from an ankle strain secured him a place on the bench.
The game started at a high tempo as the home side slightly edged the opening proceedings, with Woolford looking particularly dangerous down the flanks.
Woolford produced the best chance of the early exchanges in the 10th minute, whipping in a left-wing cross which just evaded three United players in the box.
The Iron, though, continued undeterred and came close to breaking the deadlock in the 22nd minute when Michael O'Connor's drive was saved by Scott Loach.
Jordon Mutch almost netted for Watford shortly afterwards, before Scunthorpe returned to the attack with Chris Dagnall and Thompson both coming close.
The game changed in the 32nd minute after Jenkins lunged in two-footed on Eddie Nolan and was fortunate not to receive a card.
After a few heated moments Watford seem to galvanise while the home sidewilted, with the visitors bringing on Deeney for Don Cowie in a bid to aid their ailing attack before the break.
The second half started in a slightly low-key fashion, before a power cut halted play at Glanford Park for the second time this season.
Much of the interior power and the floodlights went out in the 53rd minute, with play restarting more than 10 minutes later when the main fuse was replaced.
Scunthorpe began strongly after power was restored and were awarded a penalty after Andrew Wright was fouled in the box by Will Buckley.
O'Connor stepped up to convert the penalty but saw his effort fly over the bar and into the visiting fans.
Woolford, though, atoned for his team-mate's error by finishing well after Dagnall unlocked the Watford defence.
Scunthorpe's joy was short-lived, though, as Graham equalised within minutes, before play was delayed again by two further power cuts.
Despite referee Andy Haines considering calling the game off, the fixture recommenced after a lengthy interruption and Whichelow secured an all-important victory for the high-flying visitors.
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