Warwickshire were bowled out for 449, missing out on full batting points.
But after reaching tea on 107-1, Surrey collapsed in the final session.
Patel took 5-61, to follow his first-innings 5-62, as Surrey slumped to 177 all out, before the Bears took five balls to reach 5-0 for a 10-wicket win.
Ian Bell's men claimed the extra half hour to avoid having to come back for the fourth day.
Surrey have now failed to beat Warwickshire in their last eight home meetings.
The visitors climb to the top of Division One for the first time this season with only their second Championship victory of 2016 - and Bell's first since being appointed county captain.
After Keith Barker had earlier cracked 65 in 70 balls with 11 fours, Rory Burns led Surrey's fightback against a daunting 176-run first-innings deficit.
But after his departure for 68, the Surrey batting folded, only Worcestershire old boy Steven Davies (44) and Zafar Ansari (50) making double figures.
Patel's match figures of 10-123 were the best of his first-class career, beating the 10-163 he took in an innings victory over Sussex at Edgbaston in 2011.
With the rest of the current round of Championship games still to be completed, Surrey, who last week put in 28 days' notice to sign Durham duo Scott Borthwick and Mark Stoneman, climb 18 points clear of bottom club Hampshire.
Surrey director of cricket Alec Stewart told BBC Radio London:
"Burns and Ansari showed the way. They played tough cricket. Unfortunately, there was then a collapse. The batting is faulty.
"We bowled well. Our bowling is moving in the right direction, but our batting isn't. That is an issue.
"If you are only making scores of 270, and not batting four sessions, you are making it very hard. There are players with talent, but they are not battle-hardened cricketers."
Warwickshire spinner Jeetan Patel told BBC WM:
"It is so hard to win games in this league. Every team is really good. "There is still a long way to go. No-one is talking about being top.
"We went in at tea and the conversation was all about doing the hard yards and then eventually it will pay you back.
"We were lucky when Rory played on, then we got Zafar. All we wanted was a sniff. We knew if we got one wicket we would get two or three."
Warwickshire spinner Jeetan Patel took his second five-wicket haul of the match as the Bears went top of the County Championship by beating Surrey inside three days at Guildford.