Dugan put the home side up for the first time with seven minutes remaining when he slotted a field goal to put the hosts up 19-18 in front of 7862 spectators.
Brave Canberra were set a tough challenge early in the match, when the visitors piled two tries in the space of just seven minutes to set up a 12-0 lead.
The first would have sent a shiver up Canberra coach David Furner's spine, with Slater cruising through the Raiders' defence from close range in the third minute.
But the Raiders weren't to be denied, claiming back a heap of possession which was rewarded in the 14th minute when Shaun Fensom rocketed onto a David Shillington pass two metres out.
It swung the momentum Canberra's way and saw them score another try through Josh McCrone, who benefited when Melbourne couldn't scoop up a Terry Campese grubber.
MELBOURNE 24 (B Slater 2 C Cronk J O'Neill tries C Smith 4 goals) bt CANBERRA 19 (G Buttriss S Fensom J McCrone tries J Croker 3 goals J Dugan field goal) at Canberra Stadium. Referee: Gavin Badger, Brett Suttor. Crowd: 7,862.