Alastair Cook scored 106 to reach his sixth hundred in 12 Test innings - a remarkable statistic thought unprecedented among Englishmen - and sharing a century stand this morning with Kevin Pietersen (72).
But Strauss did not declare before midafternoon, on 335 for seven, and setting Sri Lanka a notional 343 to win in 58 overs.
Unlike in Cardiff last week, Sri Lanka cosily closed out the draw by reaching 127 for three to stay just 1-0 down in this three-match series.
Strauss told Sky Sports 1: "We thought we had a chance if we took early wickets. We knew the wicket on the fifth day at Lord's would be flat so we needed something special to happen. It died a bit of a death which is a shame.
"There are a lot of positives to take out. It was not the best Test match we have ever played but we move on to the Rose Bowl in good spirits."
England could face a tricky selection dilemma should James Anderson recover from injury with Steve Finn, who took four wickets in the first innings at Lord's, and Stuart Broad both vulnerable.
Pietersen had been the subject of some criticism but responded well.
"He played really well," said Strauss. "It was only a matter of time. He looked in pretty fine fettle."
Cook's ton - following a first innings 96 - took him within four of the England record of 22 and he told Sky Sports 1: "My form's nice and you have to cash in while it's nice.
"I was disappointed to miss out in the first innings and I was pleased to go on and get one in the second. It's been an amazing turnaround since last summer, I've got a great mentor [Graham Gooch] and we've been working as hard as ever."
Despite his score, Cook's pedestrian strike rate in a morning session in which England were looking for quick runs came in for some criticism.
But he said: "When we went out there today we were only 140 runs ahead, so if you lose a couple of wickets it's not a great position. It was about getting through that first hour and then building a partnership with Kev. We actually scored at four an over in that session - obviously I didn't do much of that scoring but you have to give credit to how Sri Lanka bowled.
Tillakaratne Dilshan, the Sri Lanka captain, was named man of the match for his first innings score of 193, although he was unable to appear in the second innings after suffering a hairline fracture of his right thumb when he was twice hit by Chris Tremlett.
His appearance at the Rose Bowl is in serious doubt with the match starting on June 16.
Dilshan told Sky Sports 1: "It is a small fracture. I will have to wait for 10-15 days and I shall see how it goes before the next Test match.
Kumar Sangakkara would replace Dilshan as captain.
Of his team's efforts at Lord's, Dilshan said: "Overall I am happy. It was all round a very good performance after the Cardiff match."
Sri Lanka 479
England 2nd Innings (overnight 149-2):
A Strauss lbw b Welegedara 0
A Cook st P Jayawardene b Herath 106
J Trott b Herath 58
K Pietersen b Herath 72
I Bell not out 57
E Morgan c Lakmal b Fernando 4
M Prior run out (P Jayawardene) 4
S Broad c P Jayawardene b Fernando 3
Extras (lb12, w1, nb18) 31
Total (7 wkts dec, 78.1 overs) 335
Did not bat: G Swann, C Tremlett, S Finn
Fall of wickets: 1-0 (Strauss), 2-117 (Trott), 3-244 (Pietersen), 4-305 (Cook), 5-312 (Morgan), 6-319 (Prior), 7-335 (Broad)
Bowling: Welegedara 10-1-50-1 (3nb); Lakmal 17-0-70-0 (7nb, 1w); Maharoof 7-0-24-0 (2nb); Fernando 201-2-92-2 (6nb); Herath 24-2-87-3;
Sri Lanka (2nd Innings):
T Paranavitana lbw b Trott 44
K Sangakkara c Morgan b Tremlett 12
M Jayawardene c Pietersen b Broad 25
T Samaraweera not out 17
P Jayawardene not out 12
Extras (b7, lb3, w6, nb1) 17
Total (3 wkts, 43 overs) 127
Did not bat: F Maharoof, R Herath, D Fernando, S Lakmal, C Welegedara, T Dilshan
Fall of wickets: 1-13 (Sangakkara), 2-66 (M Jayawardene), 3-96 (Paranavitana)
Bowling: Tremlett 9-1-31-1 (1nb); Broad 9-2-29-1 (1w); Swann 12-2-19-0; Finn 8-2-31-0 (5w); Trott 4-1-5-1; Pietersen 1-0-2-0.
Result: Match drawn; England lead 1-0