Hull KR winger Dave Hodgson led the way with two tries as the Robins earned their sixth win in seven seasons on Hull FC soil at the KC Stadium.
Rovers led 12-0 at the break, Josh Hodgson surging over, followed by Dave Hodgson's effort in the left corner.
The hosts responded within two minutes of the restart through Ben Crooks.
But Hodgson got his second, his eighth try of the season, Craig Hall adding his third kick, as well as a drop goal, before Danny Tickle's late consolation.
Rovers go four points ahead of their city rivals in the Super League table.
In the first competitive Hull derby of 2013, Rovers made light of the absence of hamstring victim Michael Dobson and a second-half injury to Travis Burns.
The Robins' 12-point first-half lead was built on the back of three crucial handling errors from the Black and Whites.
Rovers old boy Ben Galea knocked on 10 metres out from a poor Tom Briscoe offload. A penalty followed and a couple of plays later Josh Hodgson barged over from dummy-half.
Having landed the conversion, Hall then added a penalty before fast hands by Cory Paterson and Hall created the opening for winger Hodgson in the corner.
Crooks was then brought down by a reckless high tackle from Burns but, while he went unpunished, Graeme Horne was sin-binned before the break for a high tackle.
The home side, missing the sidelined Andy Lynch. failed to take advantage but, within 90 seconds of the restart, they broke wide left, Richard Whiting feeding Crooks, who dived over in the corner.
Tickle missed the conversion to leave an eight-point gap but, despite losing Burns to a shoulder injury, a great pass by Welham set up the prolific Hodgson's second try.
After stand-in scrum-half Hall converted from the touchline for an 18-4 lead, Welham then went over with 15 minutes left.
Having missed the conversion attempt, Hall's drop goal added the icing on the cake to celebrate Rovers' 100th Hull derby victory before Tickle converted his own late consolation try.
VIEW FROM THE DRESSING ROOM
Hull FC coach Peter Gentle:
"I got a wake-up call from Andy Lynch saying he had been vomiting all night.
"He wasn't able to play so the day didn't start well. But we just had no direction whatsoever. We've been looking for over 12 months to get a scrum-half in and it's proving very difficult.
"There were a lot of errors today that were unforced. We just can't find our way around the park at the moment.
"We've got Wakefield on Monday and we better come up with some answers pretty quickly."
Hull KR coach Craig Sandercock:
"It was a pretty special performance. To come through a lot of adversity like that is very pleasing. It was good to get a win without Michael Dobson.
"Michael trained really well all week and he was fine but, at the end of training on Thursday, he pulled up with that hamstring. Obviously, he won't be playing on Monday against Wigan and we'll have to assess him during the week.
"Travis Burns hurt his shoulder and he doesn't look too good for Monday either. Again we'll have to assess the damage.
"Craig Hall did very well today, as did 16 of his other mates. For Craig to jump in there without having trained in the halves all week and put in a performance like that, it's pleasing."
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