Potts's early goal gave the Seasiders the initiative in the League Two playoff final and although David Wheeler equalised for Exeter, Cullen's close-range effort midway through the second half ensured that Blackpool will only be spending a year at the fourth tier.
The triumph also caps a fine first 12 months in charge for Gary Bowyer, who can now look forward to fixtures against former side Blackburn Rovers next season.
Blackpool required just 130 seconds to open the scoring in North London as a long pass towards Cullen resulted in the forward touching the ball into the path of Brad Potts, who produced a superb first-time finish into the bottom corner from 20 yards.
The Grecians continued to have the better of the play but aside from a long-range effort from Lloyd James, they were not doing enough with the ball in and around the penalty area.
As the end of the first half drew closer, Blackpool began to improve but with five minutes remaining, Exeter were back on level terms thanks to Wheeler, who controlled a long ball before lobbing Slocombe from around 18 yards out.
Blackpool returned for the restart on the front foot with Bright Osayi-Samuel curling an effort wide of the far post and Kyle Vassell forcing a solid stop out of Christy Pym with a half-volley from 10 yards out.
They continued to make all of the running as the match progressed past the hour mark and with 25 minutes left, they went ahead through Cullen who diverted the ball into the bottom corner from close range after Potts had seen his effort saved by Pym.
Exeter had offered very little during the second half and they could have fallen further behind when Osayi-Samuel did well down the left before setting up Neil Danns, who drilled a shot over the crossbar from 20 yards.
As the game approached the final 10 minutes, Reuben Reid flashed a volley marginally wide of the far post as Exeter finally started to apply some pressure on the Blackpool goal.
However, opportunities remained few and far between and ahead of added-on time, it was Blackpool who looked most likely to net a third with Osayi-Samuel sending a shot wide of the far post and Vassell being denied by Pym from inside the penalty area.
That goal did not arrive but they successfully saw out the six minutes of added-on time to seal a memorable win in their turbulent recent history.