He found Dobson, whose long pass was taken by Welham, who handed onto Cockayne.
Dobson's decision to run the final tackle instead of kick provided the impetus for Briscoe's try as he found Chas I'Anson who offloaded to the full-back close to the line, the Australian adding the extras.
Poor Rovers play led to Matt King's try after I'Anson had the ball stolen from him by David Solomona, who passed to King, and he powered through a three-man tackle to score, with Westwood converting.
Warrington scored first after the break from a superb 40-20 kick.
The ball was then worked wide and Atkins handed off Fox and slipped the pass to Riley.
But, minutes later, Dobson's kick set up Fox, who crossed the whitewash with Dobson adding the afters.
Rovers upped the pace in the final 20 minutes and it proved fruitful when Cook crossed the line to put the home side 10 points clear.
A grubber kick through ricocheted off a defender and Cook's brilliant pick-up gave him the room to stretch for the line before Dobson converted.
England winger Fox rounded things off when he profited from an ill-advised Solomona offload and raced home to ground the ball under the posts.
Hull KR coach Justin Morgan was quick to praise winger Peter Fox:
"Peter takes a great deal of pride in how he carries the ball and I thought he did a great job," said Morgan.
"He snuffed out some of their attacks and he was very determined to have a good game.
"He is consistently doing the little things that are contributing to our performance.
"He enjoys carrying the ball out of our own end but he's a winger and they're paid to score tries."
Warrington coach Tony Smith said:
"We were second best by a long way. The game was in the balance and was looking for a team to take control. It wasn't us today.
"Hull KR looked sharp and our execution was off. We were sloppy in some of our passing and we let ourselves down a bit with some of our decision making and our discipline.
"We gifted them some penalties and didn't make them struggle. All credit to them. They got their just deserts and we got ours."
Hull Kingston Rovers: Hull K R: Briscoe, Fox, Welham, Webster, Cockayne, I'Anson, Dobson, Vella, Murrell, Clinton, Newton, Netherton, Galea, Lovegrove. Replacements: Hodgson for Webster (27), Lovegrove for Vella (21), Watts for Clinton (23),
Warrington Wolves: Mathers, Hicks, King, Atkins, Riley, Grix, Myler, Morley, Higham, Carvell, L. Anderson, Westwood, Harrison, Wood. Replacements: Bridge for Higham (28), Wood for Carvell (7), Solomona for Harrison (32), Mitchell for Wood (52).
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