England were 211 for two in reply to Sri Lanka's first innings 257, a deficit of 46 runs, at tea on the second day.
Robson was 98 not out, having put on 142 with Gary Ballance (74) for the second wicket after England captain Alastair Cook had been dismissed early in the day's play.
Ian Bell, in his 100th Test, was eight not out.
Robson was left two runs shy of a coveted century when, to the last ball before tea, he played defensively to Sri Lanka captain Angelo Mathews.
Cook was 14 not out, having survived an appeal for a catch by Kumar Sangakkara in the slips on five when replays were inconclusive.
But there was no doubt when Cook, who had added just three runs to his overnight score, felt uncertainly for the ball outside off stump and edged Dhammika Prasad to Sangakkara at first slip.
That meant in 23 innings since compiling the last of his England record 25 Test hundreds at Headingley last year, Cook had scored 585 runs at a modest average of 25.43.
But Robson, 21 not out overnight, confidently off-drove Shaminda Eranga for four in what was an important innings after his England debut at Lord's yielded scores of just one and 19.
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