Strauss fell for 48, edging champion off-spinner Muralitharan, playing his first Test at Lord's, to Sri Lanka captain Mahela Jayawardene at slip to leave England 86 for one.
Trescothick was 30 not out in his first Test since appearing against Pakistan in Lahore late in 2005 after missing the 1-1 drawn series in India earlier this year for personal reasons.
Strauss, on his Middlesex home ground, faced 97 balls including eight fours after both he and fellow left-hander Trescothick had batted serenely against Sri Lanka's seamers.
Murali, however, was an instant threat and at lunch had figures of one wicket for four runs in 3.1 overs.
Andrew Flintoff, continuing as England captain in the ongoing absence of Michael Vaughan with a knee injury, chose to bat despite the fact that no Sri Lankan batsman had made a fifty in the team's final warm-up match, a 10-wicket defeat by England A, last week.
However, in what was the earliest Test ever played in England and the home side's first on their own turf since they regained the Ashes against Australia last year, blue and sunny skies promised ideal batting conditions.
England's openers were largely untroubled against a new-ball duo lacking real menace although left-armer Chaminda Vaas and Farveez Maharoof occasionally beat the bat.
Strauss got off the mark with a four through mid-wicket off Vaas that took him past 2,000 runs in Test cricket.
Boundaries came steadily in the opening Test of this three-match series, the fifty up in 92 balls.
Sri Lanka, after 22 overs of pace bowling, brought on Muralitharan shortly before lunch.
His first ball just missed Strauss's off-stump and in his second over Trescothick, on 28, appeared fortunate to survive a confident lbw appeal, although he was well forward.
Trescothick was the only player aged over 30 in an England team where fast bowler Sajid Mahmood was making his Test debut and batsman Alastair Cook, paceman Liam Plunkett and spinner Monty Panesar were all playing in their first home Test.
Sri Lanka gave a Test debut to middle-order batsman Chamara Kapugedera.