Dobson scored his first try of the season as part of a 15-point haul
Hull KR twice surrendered a 12-point lead but still managed to stretch Harlequins' winless run to seven games.
Rovers led through tries from Jake Webster and Louis Sheriff, with Oliver Wilkes and Luke Gale levelling.
Michael Dobson and Ben Galea restored the home lead by half-time, only for Luke Dorn and Lee Mitchell to again redress the balance.
Scott Murrell put Rovers back in front, with Dobson's drop-goal and Ben Fisher's late score sealing the win.
The victory gave Justin Morgan's men their first success in three outings and provided the perfect antidote to Friday's derby defeat by Hull FC.
They took the lead when Blake Green's delayed pass put Webster over, with Sheriff marking his debut with a try after Galea and Ben Cockayne combined well on the left.
Dobson added the extras to both scores, the first of five successful kicks in an individual haul of 15 points.
Quins' first fight back began when Sheriff could not prevent Wilkes crossing - Luke Gale kicking the extras as he did five minutes later when Temata brushed off Rhys Lovegrove to score with ease.
Having being pegged back, Rovers finished the first half on top and regained their 12-point lead when Dobson took advantage of some slack defending to score his first try of the season before his 40/20 allowed Galea to dive through and score.
But again Quins fought back, first through Dorn's 50-metre run, and then with Mitchell taking Danny Ward's offload to cross.
Finally, Rovers took a lead that they would not surrender.
Murrell side-stepped into a gap with 14 minutes left on the clock, Dobson's drop-goal made Quins' task all the harder and Fisher crossed in the dying seconds.
Hull KR coach Justin Morgan: "Good Friday is always an emotional day for this club and what happens on the pitch takes a lot out of the players.
"We got out in front and we got our tails caught from behind. We then stopped their momentum and they caught our tails again.
"We were able to stop their momentum two or three times in the game. To do that on Easter Monday is a huge testament to the players' character."
Harlequins coach Rob Powell: "I thought it was a real battling performance from our blokes to get back in it at 24-24.
"We couldn't get there in the end, but we are pretty close and we're not far off putting a score on a team.
"It's not a question of lack of effort or desire and the boys are putting in 100% effort every week."
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