The Knights batsmen could not fathom the late swing Birch obtained in cloudy conditions, crashing to 106 all out in their second innings, leaving the Warriors with a target of just 94 to win the match.
Birch finished with career-best figures of eight for 30 as he claimed four of the five wickets to fall on Saturday morning after the Knights had resumed on 76 for five.
He struck for the first time in his second over of the day, having Patrick Botha (18) caught behind by Adrian McLaren to break a stubborn sixth-wicket stand of 33 with Dean Elgar.
Elgar, fresh off his maiden Test century last weekend, top-scored for the Knights with 34 before he was also caught behind off Birch four overs later.
Birch then wrapped up the Knights tail for the addition of just 20 more runs as the home side completed a dismal second-innings effort that lasted just 32.1 overs.
Quickfire innings of 48 not out by Michael Price and 44 not out by Davey Jacobs would then see the Warriors go to victory in just 13.3 overs without the loss of a wicket.
The Warriors earned 17.62 bonus points to go to 92 points overall, just 0.18 points behind the log-leading Cape Cobras.