Referee Martin Fox ruled that the Irish front row had illegally halted the drive by a rampant Warriors pack in front of their own post.
Drahm then added a superb 40-metre drop goal in the 80th minute before Irish launched a desperate all-out assault to try and retrieve the situation.
Worcester were penalised three times near their own line, with skipper Pat Sanderson receiving a yellow card for killing the ball – but they held out.
The win marked their second successive fightback after they came from behind to draw with Gloucester on the opening day of the season.
They had trailed for much of the match after winger Scott Staniforth poached two first-half tries for Irish.
Speed merchant Staniforth twice won the race to the loose ball behind the Worcester line as Irish attempted to follow up their 27-11 opening weekend win over Leeds in the Twickenham double-header.
A handling error by James Brown, Worcester’s former London Irish fly-half, presented his old club with the opening try as the ball was hacked forward and Staniforth got his hand on it first.
Fly-half Barry Everitt converted but then missed an easy penalty.
Brown’s unlucky day continued when he limped off in the 24th minute to be replaced by Drahm, the close-season signing from Northampton who promptly missed a close-range penalty bang in front of the target.
The new cavalier approach of the Exiles, whose players have pledged a more attacking game this season, is fraught with danger and it cost them a try when Worcester number eight drew Hickey and collected a poor kick in midfield and ran through some weak tackling.
Centre Rodd Penney did well to get back and haul him down from behind but Worcester kept up the momentum from the breakdown to push prop Tony Windo over for a 28th-minute try which Drahm converted to put them level at 7-7.
A minute later Irish were back in front when French international flanker Olivier Magne, making his home debut after signing during the summer, hoisted a speculative kick from the right touchline and Staniforth pounced to get his fingertips to the touchdown inches before the ball went dead.
The usually reliable Everitt, who landed only two of his five kicks, missed the conversion.
Drahm cut the lead to two points with the first of his two well-executed drop goals three minutes later with Everitt landing a penalty for Irish just before half-time.
But their opportunist approach faded in the second half as they were dragged into a forward dogfight by Worcester.
Chris Horsman, the Warriors’ prop, was sin-binned in the 46th minute for a high challenge which inflicted a facial injury on Irish flanker Kieron Dawson, who retired hurt for spell.
Then Irish full-back Delon Armitage followed him after being shown a yellow card for a blatant case of obstruction.