After resuming 41 behind on 322-7, the Bears' hopes of saving the game pretty much rested on Jonathan Trott.
But Tom Curran broke through, trapping Jeetan Patel lbw - and Trott added just 10 to his score before holing out off Mark Footitt for 151.
Chris Wright was then bowled by Curran as Warwickshire were all out for 362.
It was a good second-innings effort by the Bears, built around the 41st century of Trott's first-class career.
But they were always staring at defeat after being bowled out for 91, either side of tea on Saturday, to follow-on 363 runs in arrears - and they take just a single point from the match.
It was perhaps fitting that Footitt ended Trott's epic knock, in the eighth over of the morning, after the former England batsman had aimed an upper-cut at a short ball from the left-arm paceman, only to see it balloon into Dom Sibley's hands at third man.
That gave Footitt a match haul of eight wickets to continue the England hopeful's fine form since reaching full fitness again at the end of last season.
In Surrey's last three home Championship matches going back to last season, out of a possible 50 wickets, Footitt has taken 22 of them.
Having opened up with a 23-point haul, Surrey's next game is also at home against Lancashire, starting on Good Friday, when Warwickshire are also back in action, hosting Yorkshire at Edgbaston.
Surrey captain Gareth Batty told BBC Radio London:
"We're absolutely delighted. Both Mark Footitt and Mark Stoneman will deservedly get the plaudits for their first-innings performances, but we can get better. This is just one win.
"The fact that we were able to leave out two England players (Stuart Meaker and Zafar Ansari, plus Jason Roy, who is playing in the IPL) is a big statement. But we are not an 11-man team. It is all about the squad. It could be me sitting out the next game because people will need resting and we will always pick the right team for the surface.
"There are some seriously good teams in this division. We have now got the best domestic first-class cricket in the world. The cricket I've seen played in the last two or three years has been the best of my time in the game.
"Full credit to Trotty, who is still one of the best players in the country. It was a very good lesson for our bowlers to see how someone that good goes about his business."
Warwickshire bowling coach Alan Richardson told BBC WM:
"You just can't give sides the sort of start we gave them on the first couple of days.
"We're pleased we showed some fight after that, and Trotty was superb, but this game is a wake-up call for us.
"We bowled well on the second day, too, but overall we're disappointed. We have spoken about how we need to bounce back."
Surrey took less than an hour on the final morning at The Oval to wrap up a season-opening victory over Warwickshire by an innings and one run.