Alex McLeish's side powered to victory thanks to a double from Sotirios Kyrgiakos and further strikes from Maurice Ross, Nacho Novo and Fernando Ricksen, Well replying through David Partridge.
Ross opened the scoring just five minutes in, chipping a hesitant Gordon Marshall after latching on to a Thomas Buffel pass.
Kyrgiakos doubled the lead four minutes later, heading home following a Barry Ferguson free-kick from the right, but Well got back in the game with just 13 minutes on the clock.
Partridge powered a header past Ronald Waterreus from Steven Hammell's right-wing free-kick, having outjumped Kyrgiakos to convert.
Twelve minutes before the interval the Gers were 3-1 in front as Dutchman Ricksen curled home a free-kick from 30 yards after Stephen Craigan had brought down Novo.
Three minutes into the second half the game was over as Novo was played clear by Dado Prso's through ball and he finished confidently over the advancing Marshall.
Greek stopper Kyrgiakos had the final word four minutes from time as he met a Ricksen corner to plant a header past a helpless a helpless Marshall for his side's fifth goal.