England, in their first Test since their 5-0 Ashes drubbing in Australia and with a new-look team including three debutants, thought they'd won when Stuart Broad had Pradeep given out lbw off the penultimate ball of the match.
But the batsman reviewed Australian umpire Paul Reiffel's decision and, with technology showing an inside edge, he was reprieved.
Sri Lanka, set 390 to win after England captain Alastair Cook declared before the start of Monday's play, finished on 201 for nine.
Instead the result left the teams all-square ahead of Friday's second and final Test of the series at Headingley.
Sri Lanka appeared in little danger at 159 for two until Anderson took three wickets in 14 balls either side of tea.
First he had Mahela Jayawardene caught behind for 18, hanging out his bat.
And he then dismissed Kumar Sangakkara, who in common with Jayawardene was probably playing his last Test at Lord's, for 61 when the left-hander played on driving loosely outside off stump.
Lahiru Thirimanne never looked at ease and it was no surprise when he fell for two, edging Anderson to debutant Chris Jordan at second slip.
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