Sri Lanka's collapse allows the hosts to a seven-run lead and raises hopes of forcing a result on the final day at Lord's.
Cook sets up victory shot against Sri Lanka
The hosts, at stumps on the fourth day of the second Test, were 149 for two in their second innings — a lead of 156 — with Alastair Cook 61 not out.
Cook, under skies brightened by the Lord's floodlights, helped England recover from none for one after the exit of Strauss and together with Jonathan Trott (58) put on 117 for the second wicket.
Kevin Pietersen, despite coming in when Rangana Herath was on — left-arm spinners, including Herath, have removed him 19 times in Tests — was unbeaten on 15.
England had dismissed Sri Lanka for just 82 inside 25 overs to force an improbable innings and 14-run win in the similarly rain-interrupted first Test in Cardiff.
But here they still have 98 overs left on Tuesday's final day in which to achieve a win that would put them 2-0 up in this three-match series.
And the time remaining means a Sri Lanka victory can't be ruled out completely either.
Sri Lanka, after rain washed out all of Monday's morning session, made 479 in reply to England's first innings 486 — a deficit of just seven runs.
Strauss was then plumb lbw to left-arm seamer Chanaka Welegedara off the second ball of the innings, the skipper wasting a review before walking back to the Pavilion.
The in-form Trott, sometimes criticised for slow-scoring, prospered against an under-powered Sri Lanka seam attack to score a 66-ball fifty, completed when he punched Dilhara Fernando through the covers for his eighth four.
Former skipper Kumar Sangakkara, captaining the side as Tillakaratne Dilshan was off the field following the finger injury he'd suffered making a Sri Lankan Lord's Test-best 193 on Sunday, then opted for spin.
And it took Herath just four balls to bowl Trott, playing over the top of a dipping delivery.
Trott's exit meant Pietersen had to face Herath immediately but he played out the over with two forward defensive shots.
Cook then drove Fernando through the covers for a sixth four to go to an 86-ball fifty.
An even bigger roar came after Pietersen, advancing confidently down the pitch, drove Herath straight back over the bowler's head for the first four of his innings.
Earlier, Sri Lanka lost two wickets — Mahela Jayawardene and Thilan Samaraweera — for no runs in three balls to be 409 for six.
But wicketkeeper Prasanna Jayawardene (no relation), who made 112 in the first innings in Cardiff, hit back with a stylish 40 featuring a six and five fours and put on 57 with Herath in a lively seventh-wicket stand.
England's pace attack, wayward on Sunday in the absence of injured Ashes spearhead James Anderson, showed signs of early improvement Monday.
But a huge total of 58 extras, including 23 byes, most of which could not be blamed on wicketkeeper Matt Prior, was evidence of a lack of accuracy.
Steven Finn, recalled in place of Anderson, took four for 108 while off-spinner Graeme Swann polished off the tail with three for eight in 16 balls.
England 1st Innings: 486 all out
Sri Lanka 1st Innings (Overnight 372-3):
Tharanga Paranavitana c Strauss b Finn 65
Tillakaratne Dilshan b Finn 193
Kumar Sangakkara c Prior b Tremlett 26
Mahela Jayawardene c Cook b Finn 49
Thilan Samaraweera c Prior b Tremlett 9
Prasanna Jayawardene c Swann b Finn 40
Farveez Maharoof lbw b Broad 2
Rangana Herath st Prior b Swann 26
Dilhara Fernando c Strauss b Swann 5
Suranga Lakmal not out 0
Chanaka Welegedara c Broad b Swann 6
Extras: (25b, 23lb, 8w, 2nb) 58
TOTAL: (all out, 131.4 overs) 479
Fall of wickets: 1-207, 2-288, 3-370, 4-394, 5-394, 6-409, 7-466, 8-472 9-472, 10-479.
Bowling: Stuart Broad 32-5-125-1, Chris Tremlett 30-8-85-2 (1nb), Steven Finn 33-8-108-4 (1nb, 3w), Graeme Swann 32.4-5-101-3, Kevin Pietersen 4-0-12-0.
England (2nd Innings):
Andrew Strauss lbw b Welegedara 0
Alastair Cook not out 61
Jonathan Trott b Herath 58
Kevin Pietersen not out 15
Extras: (5lb, 1w, 9nb) 15
TOTAL: (for two wickets, 41 overs) 149
Fall of wickets: 1-0, 2-117.
Still to bat: Ian Bell, Eoin Morgan, Matthew Prior, Stuart Broad, Graeme Swann, Chris Tremlett, Steven Finn.
Bowling: Chanaka Welegedara 7-1-25-1 (1nb), Suranga Lakmal 8-0-31-0 (3nb, 1w), Farveez Maharoof 7-0-24-0 (2nb), Dilhara Fernando 11-1-47-0 (3nb), Rangana Herath 8-1-17-1.