But fellow batting great Kumar Sangakkara (59 not out) remained resolute following his first-innings 147.
With one session left in the match, Sri Lanka were 164 for three in their second innings -- needing a further 226 runs to reach their imposing victory target of 390 in a minimum of 34 overs.
It seemed the match was meandering towards a draw when Jayawardene, who in common with Sangakkara was likely playing his last Test at Lord's, was caught behind hanging his bat out to James Anderson for 18.
But given just three wickets had fallen in the day's play thus far, it was still going to take a huge effort from England's attack to force a win that would put them 1-0 up in this two-match series ahead of Friday's second Test at Headingley.
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Sangakkara, bidding to become only the second overseas player after the West Indies' George Headley in 1939 to make hundreds in both innings of a Lord's Test, appeared to hold the key to the outcome.
Indeed, should the left-hander reach three figures on Monday, he would become only the fifth player to make hundreds in four successive Test innings following his scores of 319 and 195 against Bangladesh in Chittagong in February.
That left Gary Ballance on 104 not out -- the Zimbabwe-born left-hander's maiden Test century in only his second match at this level.
The 24-year-old's innings had helped England recover from the struggling position of 121 for six on Sunday.
Cook's declaration left Sri Lanka needing to break the Lord's record for a winning fourth-innings total in a Test of 344 for one, requiring 342, set by the West Indies against England in 1984, for victory.
Although this was a fifth-day pitch, it remained a fundamentally good surface to bat on and, with little in the way of swing, England had their work cut out.