Curran weighed in with 62, coming in at number eight, backed up by first-innings half-centurion Rory Burns (49), Steven Davies (42) and nightwatchman Stuart Meaker (41), to help total 390.
That left the Bears with a daunting chase of 396 - and they began badly.
Ian Westwood went third ball to Tom Curran as the hosts closed on 2-1.
After resuming on 33-1, Surrey's lead was up to 117 by lunchtime after Meaker had batted throughout the first session for an unbeaten 29, his highest score in two years.
Former Sri Lanka left-hander Sangakkara, back from helping the Tallawahs win the Caribbean Premier League, then made the smooth transition from T20 to four-day cricket with a stylish half century, as he played first-class cricket for the first time in two months.
Sangakkara, who played for Warwickshire in 2007, shared a third-wicket stand of 60 with Meaker, who eventually batted almost three hours for his 41, before he was caught behind pushing forward to off-spinner Jeetan Patel.
Wicketkeeper Davies then passed 1,000 first-class runs for the season for a sixth time in making 42 before three sixes and five fours came in a 56-ball cameo from the younger of the two Curran brothers.
The return of Chris Wright (4-75) got rid of Davies at deep mid-wicket, and both Curran brothers caught behind by Tim Ambrose.
But, when Mark Footitt holed out to long on to leave Warwickshire with an uncomfortable two overs to face, Westwood quickly departed, lbw.
Kumar Sangakkara hit 88 against his old county and Sam Curran also made a half-century as Surrey batted most of the day against Warwickshire at Edgbaston.