James McArthur and Andros Townsend also scored for the Londoners before Marko Arnautovic grabbed a consolation goal for the Premier League's bottom team in the dying seconds.
Tomkins opened the scoring in the ninth minute when he stole in behind Jonathan Walters to toe-poke home Townsend's free kick, and three minutes later Dann made it 2-0 as he headed Jason Puncheon's corner into the back of the net.
McArthur fired the third in the second half, his shot deflecting off Geoff Cameron before flying into the net in the 72nd minute, and Townsend completed the rout by curling a bouncing shot home three minutes later.
Arnautovic netted Stoke's third league goal of the season just before the final whistle but Mark Hughes' side have now lost four straight Premier League games and conceded four goals on three occasions.
(Reporting by Philip O'Connor, editing by Alan Baldwin)
