At tea on a rain-marred fourth day, England, 2-0 up in the three-match series, were 206 for six in their second innings.
That gave them a commanding lead of 334, with no side having made more to win in the fourth innings of a Test at Lord's than the 344 for one posted by the West Indies against England in 1984 when opener Gordon Greenidge made a superb 214 not out.
The Nottinghamshire opener got closer still at Lord's on Sunday with 94 before he was lbw to Sri Lanka captain Angelo Mathews shortly before tea.
England captain Alastair Cook was 32 not out and Moeen Ali nought not out at the interval.
After rain meant there was no play before lunch, England resumed on 109 for four, a lead of 237 runs.
Hales was 41 not out and nightwatchman Steven Finn six not out.
Indian umpire S Ravi ruled in Hales's favour but the Sri Lankans reviewed, only for a marginal 'umpire's call' verdict from the Decision Review System to spare the batsman.
Eranga did succeed in having Finn lbw for seven.
Finn's exit brought in Cook who had been unable to open alongside Hales on Saturday after suffering a knee injury while fielding close in.
Nuwan Pradeep, who had taken three wickets on Saturday, knocked over Hales' off-stump with a ball that kept low when the batsman was on 58.
Replays suggested Pradeep's front foot may have been behind the crease but fielding teams are unable to challenge a no-ball call by an umpire and Hales survived.
In February, a similar incident saw Australia's Adam Voges make 239 in a Test against New Zealand in Wellington after being reprieved on seven following an incorrectly called no-ball by English umpire Richard Illingworth.
Cook, who made 85 in the first innings, had a let-off when on six he was lbw to left-arm spinner Rangana Herath but saw a review of Ravi's decision go in his favour after DRS indicated the ball had pitched outside the line of off stump.
Sri Lanka opted against a review but replays suggested a challenge would have succeeded.
Hales drove Herath for six over long-on and next ball swept him for four.
Another swept four off Herath saw Hales into the 90s.
But six runs shy of aa hundred he was squared up and hit in front of his stumps by medium-pacer Mathews.
Hales reviewed Tucker's lbw verdict, more in hope than expectation, but the DRS 'umpire's call' ruling, which showed the ball clipping leg stump, had him trudging back to the pavilion.
The 27-year-old Hales, playing his seventh Test, faced 179 balls including 10 fours and a six.
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