Quins will consider themselves unlucky after running in three tries to the Saints' two, but ultimately it was Northampton fly-half who saw his side home with three out of four kicks at goal, while the visitors were unable to add the extra points with Andy Dunne succeeding with only one of his conversion attempts.
Drahm put the hosts ahead early with two penalty kicks and after 26-minutes the Saints ran in their only try of the match after a prolonged period of pressure to bolt ahead 11-0.
Scrum-half Mark Robinson, who had been called in as a late replacement for injured England star Matt Dawson, made it all look so simple when he put flanker Darren Fox away for the try.
But Quins hit back immediately after the break when Jim Evans charged down Drahm's attempted clearance and jumped on the loose ball to score the first of his side's tries. Dunne missed the conversion.
Saints continued to pressurise the Quins defence, but they failed to stretch their lead until they were awarded a penalty try 11 minutes after the restart when the visitors were adjudged to have killed the ball. Drahm was on hand to add salt to the wound, striking the ball through the middle of the posts.
Quins picked up their second try of the afternoon three minutes from full-time, but when Dunne again pushed the ball wide it seemed they had missed out on the bonus point.
But a try by winger George Harder in injury-time – followed by Dunne's conversion – left them trailing by just one point at the final whistle.
For Northampton:Try: Penalty tryCon: DrahmPens: Drahm 2
For NEC Harlequins:Tries: Evans, Duffy, HarderCon: Dunne