Eager to put pressure on a fragile Bears batting unit, the champions chose to bowl first on a slow but decent batting pitch.
The decision looked questionable when Warwickshire were 58 without loss and then 165 for three, but was vindicated by a superb post-tea spell from Matt Dunn.
Playing his first championship game for over a year, Dunn delivered a burst of three for four in 20 balls as the home side lost five wickets for 21 runs.
The 27-year-old's work crowned a solid collective effort by a Surrey bowling unit forced into two changes due to injuries to Sam Curran (hamstring) and Conor McKerr (heel).
Dunn (three for 53) and Ryan Patel (two for 24) came in and joined forces in the final session to send the Bears into deep trouble at 186 for eight.
But the home side's freefall was halted by a spectacular counter-attack from ninth-wicket pair Norwell, who hit 58 not out from just 48 balls, and Henry Brookes (unbeaten on 35) in an unbroken stand of 89 in 16 overs to leave an engrossing day in the balance.
Warwickshire began solidly as openers Dominic Sibley (17) and Will Rhodes (39) added 58 but they fell just before lunch, both lbw to veteran spinner Gareth Batty.
The first hour of the afternoon session brought an absorbing duel between South Africa paceman Morne Morkel, playing his 472nd senior match, and Bears academy product Rob Yates, playing just his fourth.
Yates acquitted himself well but his striking of two fours in a Morkel over sparked a superb next over from the fast-bowler.
Two rapid bouncers forced the batsman back and then an equally brisk fuller-length delivery pinned him lbw for 20.
Sam Hain (47) and Adam Hose (38) retrenched to add 72 in 24 overs either side of tea but then Surrey hit back.
Hain missed an attempt to turn Dunn to leg and was lbw, while Patel hit Liam Banks' off-stump for one.
Dunn produced a superb ball to bowl Hose and won the fifth lbw decision of the innings to remove Tim Ambrose.
Former Warwickshire favourite Rikki Clarke's brilliant slip catch accounted for Jeetan Patel off the bowling of Patel but Norwell – on his home debut – launched the fightback, which included four boundaries from Morkel's first over with the new ball.