The Wolves are now on 10 points in the league table, second only to first-placed Wigan Warriors who have a superior points difference.
This was a close game in the first half, Bradford holding an 8-6 lead at the interval, but Warrington dominated the second period with five unanswered tries.
Bradford, though, re-established their lead when Paul Sykes twisted his way over the line, and Warrington attempts to regain the lead prior to half time were fruitless.
Ten minutes alter Warrington’s advantage was furthered when Atkins and Louis Anderson combined for King to cross at the corner; from the kick off King simply ran downfield sent Atkins over the line, Bridge converting.
The home side now led 20-0 but there was no let up, Michael Monaghan walking through the static Bradford defence and Bridge tagged on another goal.
Monaghan added a drop goal to the total before helping seal the victory, a chip to the in-goal area touched down by King, Bridge adding his fourth conversion.
Warrington – 33
T: Myler (32’); Atkins (51’, 63’); King (61’, 78’); Monaghan (68’)
G: Bridge (32’, 63’, 68’, 78’)
Dg: Monaghan (75’)
Bradford – 8
T: Lynch (24’); Sykes (38’)
Ref: Steve Ganson