Having made 393 in their innings, Warwicks skittled Middlesex out for 170 and forced them to follow on.
The home side were no match for Chris Woakes, Chris Martin and Darren Maddy and were all out for 167 in 52.3 overs.
Warwicks picked up 21 points from the match and are now favourites to get promoted.
In the Test match at Headingly on day two. South Africa in reply to England's 203 all out were 322-4 at the close with Prince not out on 134 and Villiers 70 not out.