Sri Lanka were 309 for four at tea on the third day, still 266 runs behind after Joe Root's maiden Test double century had taken England to 575 for nine declared in the first of this two-match series.
But now they needed just 67 more runs to make England bat again.
Left-hander Sangakkara, 36, was 121 not out but there was no third Lord's hundred on Saturday for the 37-year-old Mahela Jayawardene in what was set to be a fourth and final Test at 'the home of cricket' for both Sri Lanka greats.
Kaushal Silva was 62 not out and Sangakkara unbeaten on 32.
Play commenced in gloomy overcast conditions that led to the floodlights being switched on.
But it was pace and bounce, rather than swing, that did for gutsy opener Silva.
He'd added just one to his score when, trying to sway out of the way of a James Anderson bouncer, he didn't lower his bat, with the ball grazing the face on its way through to wicketkeeper Matt Prior.
Silva and Sangakkara put on 97 for the second wicket.
Sri Lanka, though, were 212 for two at lunch with Sangakkara 73 not out and Jayawardene unbeaten on 29.
Sangakkara then went past his previous Lord's Test-best of 78 and and the 36-year-old entered the 90 went by lofting Ali down the ground for four.
Liam Plunkett, playing his first Test in seven years, tried bowling short from around the wicket to a packed legside field but to no avail.
And when the fast bowler reverted to over-the-wicket, Sangakkara drove him down the ground for an all-run four to move on to 97.
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