A win for the Londoners would have moved them up into fifth on the table, but Healey's strike in the 67th minute condemned them to end the day in seventh place on the 2003/04 Zurich Premiership standings.
Quins fly-half Andy Dunne drew first blood with an early penalty, but his effort was soon eclipsed by a converted try from team-mate Ben Willis, who got on the end of a rampaging break from England Sevens winger Ugo Monye.
Leicester's Andy Goode chipped away at the home side's lead with two penalty goals before Louis Deacon mauled his way over Quins' tryline on the half-hour.
Dunne replied by adding his third penalty, but the Tigers went into the break with the lead after No.8 Henry Tuilagi scored on the back of some more good work by second rower Deacon. Goode put his side five points clear with the conversion.
Harlequins bounced back into the lead soon after the interval when centre Mel Deane pounced on a Dunne pass to explode through Leicester's defence, and Dunne followed up with the two extra points.
But it was Healey who was to have the last laugh – impressive scrum-half Harry Ellis managing to put Healey into the clear, with the latter cantering over the tryline to score the crucial points.
Goode left the door open by missing the conversion, and Quins came back looking desperately for the win, but the Tigers – led impeccably, as always by Martin Johnson – weathered the storm to collect four very useful away points.
For NEC Harlequins:Tries: Willis, DeaneCons: Dunne 2Pens: Dunne 2
For Leicester Tigers:Tries: Deacon, H Tuilagi, HealeyCon: GoodePens: Goode 2