Price progressed at Alexandra Palace after taking down Cullen 4-3 in a pulsating encounter on Saturday which was taken down to the wire.
He initially held a 3-0 lead in sets and blew three match darts, allowing Cullen to come back and force a decider after hitting a stunning 170 'Big Fish' finish.
The pair exchanged seven breaks of throw in the final set as Price held his nerve to hit double 20, setting up a fourth-round clash against Welsh compatriot Jonny Clayton.
"I've got a lot more in the tank, I can play a lot better than I played tonight," the 2021 world champion Price said.
"Joe made it so tough, he just kept coming back at me and I actually thought I was going to lose the game.
"I was just desperate to get over the winning line after putting myself in a tricky position, but thankfully I got the win."
Price's progression sees him meet with Clayton, who survived a surge from Northern Ireland's Daryl Gurney to also win 4-3.
"I'm not playing Jonny, I'm playing the board, just like I always do," Price added.
"I'll go up there and concentrate on my game, we'll shake hands after and may the best man win."