Keith Watson looked to have earned United the advantage in the first leg of the third qualifying round after he bundled home from close range - but Alexander Kokorin silenced the home crowd with a 25-yard strike that looped into the net off a United defender.
United had taken the lead against the run of play in the 37th minute when Willo Flood drove home from 16 yards and they created a series of chances before Igor Semshov headed the Russians level five minutes after the break.
Debutant Radoslaw Cierzniak produced a string of excellent saves but the home side kept coming back and Watson struck in the 75th minute.
United suffered narrow defeats to AEK Athens and Polish side Slask Wroclaw in their previous two campaigns and they have given themselves a fighting chance of going one better, although the second away goal leaves Dinamo favourites.
But Dinamo have already been knocked out of three competitions by Scottish sides - twice against Rangers, including the 1972 European Cup Winners' Cup final, and by Celtic in the Champions League qualifiers three years ago.