The Pistons last won on the road Jan. 18 and had come up empty in 10 straight visits to Salt Lake City before Monday's wire-to-wire drubbing.
The Jazz fell for the seventh time in the last eight games, despite Gordon Hayward's 32-point performance.
The Pistons inched out to a 26-23 lead over the opening 12 minutes and stepped on the gas when they doubled-up the Jazz in the second by a 34-17 margin.
Drummond took over in the middle stages, scoring inside on four of five possessions -- three coming off alley-oops -- on a personal 8-2 run that stretched Detroit's lead out to 44-33.
Kentavious Caldwell-Pope's 3-pointer in the closing seconds sent the Pistons into halftime with a healthy 60-40 cushion as the Pistons wound up connecting on 65 percent (13-of-20) of their shots in the frame.
Monroe's jumper 4-and-a-half minutes into the third made it a 25-point game, and though the Jazz scored 13 of the next 14 points, they were still in the hole entering the fourth.
The closest the hosts got in the fourth was 12 -- 88-76 -- but Detroit ripped off an 11-3 run to build back its 20-point lead and cruise to the finish line.