North End striker Gallagher struck twice in the final 19 minutes as the two-time winners, who face Walsall in the home leg of the Johnstone’s Paint Trophy northern final on Wednesday, continued their fine cup form this season.
Gary Hooper squandered a great chance to put Norwich ahead in the 32nd minute and Bradley Johnson was denied a one-on-one with keeper Thorsten Stuckmann by a superb saving tackle from Bailey Wright on the hour-mark.
Preston almost went in front after 62 minutes but David Buchanan’s cross was headed over by Andy Little from 12 yards.
However, it mattered little as Gallagher slotted the ball under Canaries goalkeeper John Ruddy from 12 yards on 71 minutes after a Jack King knock-down.
And North End sealed the win on 85 minutes as Kyel Reid slipped in Gallagher who calmly finished past Ruddy from 10 yards.