The home side, in reply, reached 36 for nought at the close of play at Headingley with captain Alastair Cook unbeaten on 14 and Sam Robson 21 not out.
Broad became the fourth player in history to register two test hat-tricks and neither the all-rounder, his team mates nor the crowd realised it because the first wicket came from the last ball of one over and the second and third arrived from the first and second delivery of the next.
The fast bowler steamed in for his third ball with avociferous crowd roaring him on, seemingly unaware he had already achieved three wickets in successive deliveries.
Plunkett also took two wickets in two balls earlier as Sri Lanka, who were put in to bat at the start of play, failed to make the most of a fortunate innings of 79 from the in-form Kumar Sangakkara.
"I had no idea it was a hat-trick," Broad told Sky Sports television. "I was listening to the guy that came on the tannoy and I was thinking 'why is he mentioning the last ball of the over before?', and it clicked it might be a hat-trick.
"It was an excellent bowling performance all round. There's always pressure when you win the toss and bowl, you sort of think it should be 200 all out but we're pleased with 257."
Broad, who also snatched three wickets in three balls against India in 2011, joins Australian pair Hugh Trumble and Jimmy Matthews and Pakistan's Wasim Akram as the only players to pick up two test hat-tricks.
First-test centurion Sangakkara was Broad's first victim when his loose drive was expertly caught in the gully by Ian Bell, playing in his 100th test, to leave Sri Lanka on 228 for six.
Plunkett then removed Dhamikka Prasad for a duck after he gloved a bouncer to keeper Matt Prior.
Broad came back for the next over and had Dinesh Chandimal snapped up at slip by Cook for 45 before Shaminda Eranga feathered an edge through to Prior for another duck.
Sri Lanka had lost four wickets for one run in nine balls but England were frustrated by a last-wicket partnership of 28 until Plunkett ended the innings by removing Nuwan Pradeep for 13.
AWKWARD THROW
Sangakkara, going in at number three, was almost run out for a duck earlier when Prior failed to gather an awkward throw from Broad and was unable to break the stumps cleanly with the left-hander out of his crease.
The Sri Lankan maestro continued to lead a charmed life and was dropped by Prior on 27 before Moeen Ali also put him down on 57.
Mahela Jayawardene (22) was the second of Plunkett's victims when he edged the Yorkshire paceman, playing on his home ground, to Chris Jordan who took a good low catch.
Lahiru Thirimanne then fell first ball, fending a fierce delivery from Plunkett to Robson at short leg, before Sri Lanka captain Angelo Mathews survived the hat-trick bid.
Sri Lanka were in big trouble at 161 for five when Mathews was dismissed by James Anderson for 26 but Sangakkara and the recalled Chandimal steadied the ship, sharing a partnership of 67 for the sixth wicket.
Broad, however, dramatically swung the game back in England's favour with his hat-trick.
"I've got the match ball but I'll cut a bit off the seam for him," said Plunkett. "He didn't realise he had even got it, he was at mid-off and he chucked me the ball when the announcement came.
"I don't think anybody realised. Nobody really thought about it but a hat-trick's a hat-trick so congratulations to him."
Plunkett was brought back into the fold for the first test against Sri Lanka at Lord's after a seven-year absence.
"Eighteen months ago I'd have never dreamt of this again, to get back in the test team," he said. "I think it's important for me to show I can bowl at test level and I'm happy I did that.
"It would have been nice to get a hat-trick but I was just trying to smash the top of the stumps. Cook made the right decision and we'd take 257 any day."
The first test of the two-match series ended in a draw.
(Editing by Tony Jimenez)
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