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 Strauss's exit and together with Jonathan Trott (58) put on 117 for the second wicket.
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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.
Fast bowler Steven Finn, whose return of four for 108 saw the 22-year-old, in his 12th match at