England in their second innings were 109 for four at the close of the third day, a healthy but not impregnable lead of 237 runs.
Alex Hales was 41 not out, having been dropped on 19 and 39, with nightwatchman Steven Finn, missed on one, unbeaten on six.
England, 2-0 up in the three-match series, were rocking at 50 for three on Saturday before first-innings century-maker Jonny Bairstow helped Hales add 52 for the fourth wicket.
But he was dropped on 19 when Dimuth Karunaratne failed to hold a one-handed slip chance off paceman Pradeep.
However, it wasn't long before Pradeep had Nick Compton out for 19 instead.
Compton had been promoted to open in place of England captain Alastair Cook, after the left-hander suffered a knee injury fielding close in earlier on Saturday.
By his own admission, Compton needed a big score to secure his place.
But on his Middlesex home ground and one where a stand is named after his grandfather Denis Compton -- one of England's greatest batsmen -- he was caught behind tamely pushing at Shaminda Eranga.
Joe Root (four) was bowled by an absolute shooter from Pradeep.
New batsman James Vince, like Compton, was in need of runs, albeit this was the Hampshire batsman's maiden Test series.
But Vince was out for a golden duck as he deliberately played no shot to a Pradeep delivery that nipped back down the Lord's slope and knocked over his off stump.
Bairstow, in at 84 for four in the first innings before making a Test-best 167 not out, survived the hat-trick.
But on 32 he too was clean bowled by Pradeep.
Worse followed for Pradeep next ball when Chandimal inexplicably failed to move for a Finn edge.
Pradeep ended the day with three for 25 in nine overs but his figures could easily have been better.
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