Saif Zaib stood out with the bat for Northamptonshire but Surrey finished a sun-filled Day 2 with the upper hand.
Clarke, who will retire just before turning 40 later this month, took three for 34 as the home side were bowled out for 171 at Wantage Road, a first-innings deficit of 81.
Luke Procter struck back by dismissing both Surrey openers second time around, but Hashim Amla’s unbeaten 26 steered his side to 102 for four and an overall lead of 183 at stumps.
Surrey’s director of cricket Alec Stewart and Northamptonshire coach David Ripley took turns officiating at square leg after umpire Russell Warren was taken ill, with Martin Saggers later drafted in to replace him.
Resuming 245 behind, Northamptonshire progressed steadily to 66 for the loss of Ricardo Vasconcelos, whose loose shot to Jordan Clark sailed into the hands of deep square leg.
But Reece Topley, having beaten the bat several times during a brief opening spell, gained greater reward in his second, slanting the ball in to have Emilio Gay caught behind and repeating the trick with Procter, taken at second slip in his next over.
Northamptonshire were suddenly in trouble at 73 for four when Gus Atkinson brought one back to hit Rob Keogh’s off stump, prompting Adam Rossington to launch an immediate, albeit controlled counter-attack.
The captain hammered Atkinson for two cover boundaries – only to be trapped leg before by Clarke straight after lunch as the procession of wickets gathered pace.
Saif Zaib was the only batsman to offer prolonged resistance, registering just three scoring shots in his first hour at the crease and eventually emerging from his shell when he started to run out of partners.
The left-hander took 10 off one Atkinson over and hooked Clarke sweetly to the boundary but when he attempted it again, a bottom edge onto the stumps ended his knock of 37.
Cameron Steel’s maiden first-class wicket for Surrey, an lbw decision in his favour against Ben Sanderson, soon wrapped up the innings and ensured he and Ryan Patel were strapping their pads on at tea.
The opening pair added 38 before Procter picked up both in quick succession, while Laurie Evans became teenage debutant James Sales’ first senior scalp when he was bowled by a ball that moved away to beat the outside edge and find the top of off.
Surrey also lost first-innings centurion Jamie Smith, lbw to Simon Kerrigan, but Will Jacks kept Amla company until the close.