Steve Basham repaid the faith shown in him by new Hatters boss Richard Money, heading home the opener from Adam Newton's cross in the fourth minute on his first start since August.
Newton's clever lob then doubled the lead for the Blue Square Premier outfit in the 20th minute before Basham grabbed his second eight minutes later with a superb curling shot from the edge of the box.
But Dale roared back and Dawson deflected Rory McArdle's header past Hatters goalkeeper Shane Gore in the 58th minute to loosen the home side's grip on the game. Then Thompson crashed home Tom Kennedy's cross with just three minutes remaining to set up a thrilling finale.
Tom Craddock could still have sealed it for Luton, but Thompson made the Hatters pay dearly for the miss with a dramatic last-gasp header to earn the League Two high-fliers another shot at the Hatters.