Tom Cannon sealed just a fourth win of the season for the Blades late on to bring Chris Wilder's men within one point of climbing out of the relegation zone.
Winless against United since 2012, the defeat compounded a miserable season for Wednesday, who are set to drop to the third tier of English football.
They remain on minus four points at the bottom of the table after receiving a 12-point deduction for entering administration last month.
Campbell, who also scored the only goal in this fixture last season, opened the scoring on 11 minutes when he was set free by Callum O'Hare to fire past Ethan Horvath.
Just three minutes into the second period, Campbell doubled his tally with a cool finish before Cannon came off the bench to net just his second goal in 30 appearances for the Blades.