Pradeep negotiated the final five deliveries, including successfully reviewing an lbw decision on the penultimate ball and edging the last just short of slip, to seal the draw as Sri Lanka finished on 201 for nine ahead of the second and final test starting on Friday in Leeds.
"When you get so close and then you get the decision that wins you the test overturned, it's hard to take, and that last nick," England captain Alastair Cook told a news conference.
"But that's what test match cricket is about, going all the way down to the wire, both sides leaving everything out there."
Anderson had got rid of Mahela Jayawardene (18), Kumar Sangakkara (61) and Lahiru Thirimanne (2) in the space of 14 balls either side of tea, before returning with the new ball to remove captain and last recognized batman Angelo Mathews (18).
Cook had set Sri Lanka 390 to win in 90 overs after declaring England's second innings on their overnight score of 267 for eight, although taking 10 wickets on a lifeless wicket was always going to be a tall order for his four-strong seam attack.
While they could not quite get the job done, the performance, which included centuries by Joe Root and Gary Ballance, will give hope to an England side looking to move on from the controversial end to Kevin Pietersen's international career and the 5-0 Ashes thrashing in Australia.
"I think we've made quite a lot of the running in this test match, to lose the toss and be put in and get 580 you give yourself a great chance in the game," Cook added.
"We were ahead in the game a lot, we can take a lot from that, we scored at a really good rate, a lot of the guys can take a lot of credit for that."
Sri Lanka had half an eye on an outrageous attempt to win the match and made steady progress in their second innings, losing opener Dimuth Karunaratne for 16 before lunch.
Sangakkara and Kaushal Silva put on 98 for the second wicket before the latter fell for 57, his second half-century of the match.
SOLID BASE
The diminutive Silva held his nerve in the face of some well directed bouncers from Liam Plunkett to add to his first-innings 63, before gloving Chris Jordan through to keeper Matt Prior down the leg side.
Jayawardene also came in for a short-pitched examination from Plunkett but eventually went to Anderson as he feathered an edge through to Prior, leaving the tourists 266 behind at tea with seven wickets left.
With 34 overs left in the test, Sri Lanka were looking towards Sangakkara to provide a solid base, but Anderson had other ideas.
After passing 50 for the sixth straight innings and becoming the seventh top run scorer in tests, Sangakkara chopped the ball onto his stumps from Anderson in the third over after the interval.
Thirimanne, who failed to look at ease in his two appearances at the crease, edged Anderson to Jordan in the slips four balls later to raise home hopes of an amazing end to the test.
The visitors were teetering at 170 for five, but captain Mathews and wicketkeeper Prasanna Jayawardene, who may be ruled out of the second test due to bruising on his hand, resisted the fired-up English attack for just over 20 overs, taking a number of blows to the body in the process.
The put on 24 runs in 121 balls before Prasanna went lbw to Jordan and then Nuwan Kulasekara fell in similar fashion to Broad soon afterwards.
Mathews later edged to Cook in the slips to give Anderson his fourth wicket of a remarkable effort and the Lord's crowd suddenly believed a famous victory could be in sight.
Broad dismissed Rangana Herath with the first ball of the final over, although the batsman looked to have taken his hand off the bat handle, but Pradeep survived the final five deliveries to avoid any wrangling.
"All he said to me was don't worry I'm going to hold on," Sri Lanka captain Mathews said of Pradeep, who took a nasty blow on the shoulder in the first innings.
"It was a good effort at the end, but after tea we were not really relaxed," he admitted.
(Editing by Mark Meadows; mark.meadows@thomsonreuters.com; +44 20 7542 7933; Reuters Messaging:; mark.meadows.reuters.com@reuters.net; To sign up for our Global Sports Forum chatroom, click on https://forms.thomsonreuters.com/global_sports_forum)
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