England were 54 for two in reply to Sri Lanka's first innings 184, a deficit of 130 runs, at lunch on the third day.
Alastair Cook and Kevin Pietersen were both 22 not out after England were reduced to 14 for two after the early exits of Strauss and Jonathan Trott.
Strauss managed just three runs before he opened the face against left-arm seamer Welegedara to a ball he could have left and gave a simple catch to Tharanga Paranavitana at first slip.
Strauss's exit was the third time in as many innings he'd been dismissed by Welegedara after twice falling lbw for four and nought in the drawn second Test at Lord's.
It left Strauss with just 27 runs for the series and meant the 34-year-old had scored just one hundred in his last 35 Test innings -- 110 in the drawn Ashes opener against Australia in Brisbane in November.
But Cook -- playing his 66th consecutive Test and so breaking the England record shared by Alan Knott and Ian Botham -- drove Welegedara off the back foot for a superb boundary.
Suranga Lakmal though had Trott out for just four after a ball angled across the right-hander took the outside edge on its way to wicketkeeper Prasanna Jayawardene.
Pietersen got off the mark with a straight driven four off Lakmal and later used just the one hand in playing a similar shot for another boundary off the same bowler.
Although troubled by left-arm spinner Rangana Herath in this series, Pietersen went down the pitch to drive the first ball he faced from him for four in the last over before lunch.
Sri Lanka, 177 for nine overnight, added just seven more runs.
Welegedara was last man out, caught driving to Eoin Morgan at cover to give the wayward Stuart Broad his only wicket of the innings.
Dilhara Fernando was a Test-best 39 not out, runs that looked increasibly valuable as England struggled in reply.
England fast bowler Chris Tremlett, who until his move to Surrey last season played at the Rose Bowl for Hampshire, took a Test-best six for 48 in 20 overs.
England lead this series 1-0 after an innings and 14-run win in the first Test in Cardiff.
Victory here will see them go joint second in the ICC Test rankings alongside South Africa with only India, their next opponents, above them.
But England's prospects may be blighted by rain, which having wiped out 120 overs from the first two days of this match and delayed Saturday's start returned at lunchtime, as much as by Sri Lanka.
