Leeds captain Kevin Sinfield kicked a late drop goal to give his side a hard-fought victory against Hull KR at Craven Park.
Graeme Horne gave Rovers an early lead before Danny McGuire levelled, but further tries from Josh Hodgson and Dobson sent the Robins 18-6 clear.
Tries from Ryan Hall, McGuire and Carl Ablett put Leeds ahead before Louis Sheriff restored Rovers' lead.
Sinfield then kicked a penalty before his single-pointer secured the win.
Early Rovers' pressure paid dividends when Horne scored in the fourth minute.
Super League facts and figures
Record-breaking Kevin Sinfield has now kicked 98 goals this season to stay top of the Super League list
Hull KR's top try scorers are David Hodgson (14) and Kris Welham (13)
Danny McGuire's brace takes him to 18 for the season - two behind Rhinos top scorer Ryan Hall
Leeds were quickly back on level terms when McGuire was put clear down the middle, but hooker Josh Hodgson restored the home team's lead when he darted over after Rhinos forward Jamie Jones-Buchanan conceded a penalty.
Ablett then had what appeared to be a good try ruled out by the touch judge before Dobson extended the Robins' lead, carving his way through the Leeds defence and selling Rhinos full-back Zak Hardaker a dummy before crossing.
With Dobson goaling all three tries Rovers took an 18-6 lead into the break, but Leeds were soon back within two points when winger Hall and the influential McGuire both crossed early in the second half.
After McGuire's 200th Super League try, his side then hit the front when Ablett twisted his way over from close range.
That made it three tries in just 11 minutes for the visitors, but Hull KR regained the lead when Scott Murrell set up full-back Sheriff for a try, Dobson kicking the goal.
Sinfield then tied the scores 24-24 when he kicked a penalty (given for reefing the ball out of the tackle) before nosing his side in front with a drop-goal seven minutes from time.
VIEW FROM THE DRESSING ROOM
Hull KR coach Craig Sandercock told BBC Radio Humberside:
"It's disappointing, I thought we deserved to win that game. But that's rugby league for you, sometimes you're unfortunate and you don't get what you deserve.
"We thought we were in with a good shot against a full-strength Leeds side. To come so close and yet so far is very disappointing.
"Leeds don't have the myriad of international players for nothing. They are a class side and if you give them a sniff, they will take it.
"They got the jump on us at the start of the second half. I'm disappointed in the overall performance of the referee and we'll leave it at that."
coach Brian McDermott:
"We weren't good enough in the first half. Hull KR played really well and defended their line whereas we didn't.
"We changed some things at half-time and the change of attitude was the main thing. It was more from the players at half-time as they took charge of things.
"The second half encouraged me but we should have done that in the first half. We just missed too many tackles on our goal line, plain and simple. There's nothing technical in that."
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