Playing in only his 13th first-class match, Curran's mid-afternoon burst of three for two in 18 balls left the home side rocking.
Replying to Surrey's 252, the Bears made 247 after a 95-run stand from Keith Barker (62) and Ian Bell (43).
Having led by five runs after the first innings, Surrey then closed on 33-1.
Warwickshire had begun the afternoon decently placed at 91-3, but Curran had Laurie Evans and Tim Ambrose superbly caught by Rory Burns in the gully before trapping Surrey old boy Rikki Clarke lbw with a swinging yorker.
Former Bears skipper Ian Westwood, on his recall as opener after being dropped following early season struggles, went just four runs short of a first Championship half-century.
After adding 40 in 15 overs with Jonathan Trott, both fell in the space of nine balls from Stuart Meaker.
Warwickshire all-rounder Keith Barker told BBC WM:
"The game is in the balance. We have managed to get ourselves back in the game. It would have been better if we had been able to go past them but it is going to be a big day for us tomorrow.
"There are runs to be scored there. You have just got to stay out there long enough for the bad balls to come. It may take a while but you have got to latch on to them if you get the chance.
"There is still a bit of movement in the pitch and we have just got to squeeze them tomorrow and put them under pressure. The pitch might deteriorate a little bit so that will give Jeets something to work with.
"It is effectively a one-innings game so if we can get some wickets tomorrow it will change the game massively."
Surrey bowling coach Stuart Barnes told BBC London 94.9:
"We took 10 wickets in 78 overs, which was the goal at the start of the day. We were not 100 per cent happy with the first session, but we put it right.
"Sam Curran fully deserved his five wickets on the back of a really good first spell, which he followed up with a great second spell after lunch.
"We talked about using the wicket and surface, which has been offering a bit of nip and we were able to swing the ball all day, which helped."
Surrey teenager Sam Curran claimed a career-best 5-44 to help his side fight back on the second day against Warwickshire at Edgbaston.