Campbell slotted home in first-half stoppage-time to earn City a 1-0 away victory over Elgin.
The hosts came close to snatching a last-gasp equaliser but Callum Antell produced an excellent stop from Kane Hester.
Darryl Duffy twice came close for the hosts in the first half and Jaime Wilson had a goal disallowed for offside for Dumbarton, who dropped into the play-offs on Thursday night following Clyde's win over East Fife.
Stranraer striker Thomas Orr and Dumbarton forward Rabin Omar came close in a bright start to the second half and there were further chances.
Both Connor McManus and Orr could not get a touch to a ball that flashed in front of the Sons goal and Jamie Hammill cleared off the line from Robert Jones at the other end.