The storm had already claimed the lives of at least two people in China's Hainan and 16 people in the Philippines
Sri Lanka were skittled out for just 82 in their second innings, with fast bowler Tremlett taking the first three wickets to fall on his way to a return of four for 40 in 10 overs on the final day.
Off-spinner Graeme Swann followed up with four wickets for 16 runs in seven overs as Sri Lanka’s dramatic collapse continued.
Stuart Broad finished off the match with two wickets in three balls, the tourists second innings ending in the 25th over in under two dramatic hours.
England’s victory in their first Test since a 3-1 Ashes triumph on Australia was achieved despite rain, which fell throughout this match, washing out nearly a whole day’s play in total.
And the win was achieved despite England being without James Anderson, who was ruled out from bowling in the remainder of this match on Saturday with a side strain.
Tremlett struck sixth ball to have Tharanga Paranavitana well caught low down at first slip by England captain Andrew Strauss.
But it was all the 6ft 7in Surrey paceman’s own work when he caught and bowled Sri Lanka skipper Tillakaratne Dilshan after a review confirmed the batsman had gloved a lifting delivery back to the bowler.
The experienced duo of Kumar Sangakkara and Mahela Jayawardene hung around until tea.
But both former captains fell shortly afterwards.
The hostile Tremlett removed Jayawardene with the aid of a Strauss slip catch before Thilan Samaraweera played on to Swann for nought.
Sangakkara was then held by Strauss off Swann and 43 for five soon became 43 for seven.
Prasanna Jayawardene, who had defied England with a century in Sri Lanka’s first innings 400, fell for just three when he was given out on review, after gloving an attempted hook off Tremlett through to wicket-keeper Matt Prior.
England then reduced Sri Lanka to 52 for eight when Rangana Herath, swinging wildly, was lbw to Swann before Broad wrapped things up.
There seemed little chance of England forcing a victory after rain washed out the whole of Monday’s morning session and prevented play from resuming until 3.00pm local time (1400GMT).
England then added just five runs to their overnight total.
Strauss declared on 496 for five as soon as Ian Bell, who resumed on 98, completed a 159-ball hundred featuring a six and 11 fours.
Bell’s was the third century of an England innings that saw his Warwickshire team-mate Jonathan Trott score 203 and opener Alastair Cook 133.
England will now head to Strauss’s Lord’s home ground, where the second Test starts on Friday, 1-0 up in this three-match series.
The storm had already claimed the lives of at least two people in China's Hainan and 16 people in the Philippines
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