Sri Lanka, following on, were 116 for seven at tea on a rain-interrupted third day -- still 91 runs shy of making England bat again.
Anderson, England's all-time leading Test wicket-taker, had figures of four for 28 to add to his return of five for 16 in Sri Lanka's meagre first innings 91.
His Saturday haul included Kusal Mendis, who rode his luck to make 53 -- an innings where he gave four chances.
At tea, Lahiru Thirimanne was 15 not out and Dushmantha Chameera nought not out, with more bad weather seemingly Sri Lanka's lone hope of avoiding going 1-0 down in this three-Test series.
The overcast conditions that greeted Sri Lanka were similar to those in which they collapsed on Friday when they were undone by the new ball-pairing of Anderson and Stuart Broad (four for 21), ably supported by wicket-keeper Jonathan Bairstow, who held five catches to add to the 140 he made on his Yorkshire home ground in England's first innings 298.
Dimuth Karunaratne avoided a pair on Saturday but fell for seven when caught head-high by Bairstow after the left-hander got an outside edge to a near-unplayable Anderson ball that bounced and cut away off the seam.
Either side of the wickets, Mendis, who like Karunaratne made a first-innings duck, drove both Anderson and fast bowler Steven Finn for well-struck fours.
Mendis was reprieved again on 29 when he edged Finn only
for a diving Bairstow, in a rare blemish this match, to drop the low one-handed catch.
And off what turned out to be the last ball before lunch, Mendis should have been out for 47 when he edged a seemingly comfortable chance off Broad to third slip only for Vince to again put him down.
Rain stopped play for more than two hours and when the match resumed, England captain Alastair Cook brought on off-spinner Moeen Ali.
But Cook's gamble was rewarded two balls later when Dinesh Chandimal chopped an intended cut onto his stumps.
Mendis, whose innings was a mix of the streaky and the stylish, then clipped Anderson through midwicket for a boundary that saw him complete a 62-ball fifty including 10 fours.
Sri Lanka captain Angelo Mathews made a Test-best 160 when his side won at Headingley two years ago to claim their first series win in England and he top-scored for the tourists with 34 on Friday.
And 93 for four became 93 for five after Mendis's luck ran out when, trying to leave an Anderson delivery, he deflected the ball onto his stumps.
Wickets continued to fall and, one ball after he had been hit on the elbow by Finn, Rangana Herath chipped the paceman to Broad at mid-off.
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