Durham 401 (Jennings 201*, Footitt 5-90) and 213 for 8 (Stoneman 92, Clark 54, S Curran 6-51) lead Surrey 367 (Roy 120, Onions 5-90) by 247 runs
His sensational burst of four wickets in seven balls rocked Durham before Graham Clark counter-attacked with a 53-ball half-century.
Coming in at 83 for 5, Clark put on 109 with Stoneman before Curran returned to have them both lbw when playing to leg.
Durham were 213 for 8, leading by 247, when the murk which had forced the floodlights to be kept on all day was deemed to have got worse.
Play was halted with 22 overs still to be bowled, but there is still every prospect of an intriguing final day with Durham needing a win to ensure a 12th successive season in Division One.
Stoneman frustrated next season's employees as he built steadily on a rapid start, passing 1,000 championship runs for the fourth successive season.
He hit three fours in five balls from Mark Footitt early in his innings, the third being pulled just out of mid-on's reach. After that he throttled back, reaching 50 off 82 balls before slowing further.
The other Surrey signing, Scott Borthwick, remained 33 short of 1,000 when he took 18 balls to get off the mark then fell for 12 when he became Curran's first victim with the score on 77 for 1.
Wicketless in the first innings, Curran returned for a second spell after being given only three overs in his first.
Borthwick pushed forward to his second ball and edged to Ben Foakes. The next was slanted across right-hander Jack Burnham, who also edged to the wicketkeeper, then Ben Stokes was able to watch the hat-trick ball pass harmlessly by.
But in trying to withdraw his bat from the next he continued his disappointing match by edging to Kumar Sangakkara at first slip.
Paul Collingwood ran a four to third man but in Curran's next over could only steer a ball rising towards his chest to third slip. Three balls later Curran could have had another when Jason Roy got his left hand to an edge by Stoneman on 48.
But Curran began to stray, helping Clark off to a confident start. In nine previous championship innings, he had a top score of 36 and three ducks. But he is known as a powerful striker and on 31 he pulled Footitt behind square for a big six.
Two more powerful pulls off Tom Curran followed before Surrey turned to Zafar Ansari. Clark lofted the left-arm spinner over mid-on for his eighth four to reach 50, only to depart for 54 straight after tea.
Stoneman followed six overs later and Stuart Poynter made 19 before edging a drive at Footitt to first slip just before a halt was called.