Jason Scotland worked his way into the Blackpool danger area twice in the opening minutes, but it was Tom Butler who brought the first save from Blackpool keeper Paul Rachubka after four minutes, with a shot from 15 yards out.
But it was the visitors who opened the scoring in the 13th minute when Brett Ormerod picked up a loose clearance and slipped a pass out to Campbell, who beat Dorus de Vries in the home goal with a rising drive.
Jordi Gomez and Scotland went close as the Swans tried to get back on level terms, but the Seasiders defended in numbers.
Alex John-Baptiste missed a golden opportunity to double his side's lead when he headed a David Vaughan corner wide when well placed.
Campbell did get the ball in the net on the half hour, but was flagged for offside.
Butler and Andrea Orlandi had plenty of ball down the wings, but too often they tried to do too much and the visitors' defence were able to clear.
Swansea made a change at the interval with Joe Allen replacing Leon Britton, while Gorka Pintado went on after an hour, but still the Swans struggled to make headway against a resolute Blackpool rearguard.
Scotland was brought down by Keith Southern in the 66th minute just outside the area, but the free-kick from Gomez was deflected narrowly wide.
Alan Tate came to the rescue in the closing minutes by heading clear as Roy O'Donovan waited to meet a Vaughan cross.
Blackpool were then given a free-kick outside the area when Charlie Adam had clearly been fouled by Alberto Serran two yards inside the box, but the set-piece was blocked and the Swans went back on the attack.
In the closing minutes Mark Gower brought a fine save from Rachubka at the near post and the ball was cleared to safety.