Swansea kept pace with the Sky Bet Championship leaders with a hard-fought 1-0 victory over Rotherham.
Paul Warne's side may have more points at this stage of the season than any of their previous three Championship campaigns, but after claiming only six points from their last six games, they only sit three points clear of the relegation zone.
Cullen was tasked with providing the goals in place of top scorer Andre Ayew, who suffered a minor hamstring injury on Ghana international duty.
The first clear-cut opportunity of the afternoon fell to Cullen after a bright start by Cooper's men.
The 21-year-old rose unchallenged at the front post to meet Grimes' inswinging corner only to see Blackman dive to his left to make an athletic reaction save.
But the Millers goalkeeper had a moment to forget at the next Swansea set-piece after 27 minutes.
The Chelsea loanee rose to punch Connor Roberts' corner clear, but he instead found Grimes waiting on the edge of the box. The Swansea skipper fired a half-volley towards goal and Blackman could not find his footing in time to prevent the ball from hitting the back of the net.
Despite dominating the rest of the first half, Cooper's men had to settle for a one-goal advantage at the interval.
The Rotherham goalkeeper redeemed his earlier effort after 63 minutes to keep out a Dhanda free-kick that looked destined for the bottom corner from 25 yards out.
Rotherham looked to be growing into the game, but it took a crucial intervention from Wes Harding to prevent the hosts from doubling their lead when he got his body in the way of Jake Bidwell's close-range strike to prevent a certain goal.
Warne's men were incredulous five minutes from time after Ryan Bennett looked to have committed a handball on his own goal line after Matt Smith headed Vassell's cross towards goal.
But their protests were waved away by referee Matthew Donohue and it ended 1-0 after a barrage of late Rotherham pressure.