Sri Lanka's batsmen had no answer to an attack which moved the ball off the seam to provide a conveyor-belt of catches for keeper Jonny Bairstow, who took four, and England's close fielders.
Chris Woakes, who recently took nine wickets in an innings for Warwickshire, led the way in helpful conditions with a particularly unsettling spell to take 3-9, including the wickets of Kusal Mendis and Milinda Siriwadana in one over.
Stuart Broad also took three wickets, two coming late on as Sri Lanka's resistance broke.
England's bowlers could not have had a better platform to launch their attack than Moeen's 207-ball innings, which was his best in tests and his second century against Sri Lanka.
At times watchful and then belligerent, Moeen became only the fifth England number seven to reach three figures in tests after being promoted one place in the order to cover for the injured Ben Stokes.
The Worcestershire all-rounder moved on from his overnight 28 and made the most of the three lives the tourists gave him on Saturday.
Support came from Woakes, who shared a stand of 92, and Steven Finn, who scored just 10 of their ninth-wicket stand of 72. His wicket provided spinner Rangana Herath with momentary cause for celebration as he became only the third Sri Lankan bowler to take 300 test wickets.
Moeen reached his 150 with a huge six before Cook declared midway through the second session.
For Sri Lanka to get anywhere near such an imposing total was a huge ask and James Anderson, who had taken 10 wickets in the first test, quickly removed opener Dimuth Karunaratne with a ripper that bowled him round his legs.
Broad followed up by having Kaushal Silva taken behind by Bairstow and the keeper took further catches to remove Dinesh Chandimal off Anderson and Angelo Mathews and Siriwadana off Woakes.
Broad broke Herath's late resistance via another sharp Anderson catch and removed Shaminda Eranga as England, who won the first test in the three-test series by an innings and 88 runs, stayed on course for another crushing win.
Lahiru Thirimanne (11) and Suranga Lakmal (0) were not out at the close.
(Reporting by Neil Robinson, editing by Ed Osmond)
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