Robson, who made his debut in the first-test draw at Lord's last week, was the model of patience as he stroked a risk-free 98 not out.
The other not out batsman was Ian Bell who was on eight on his 100th test appearance.
Sri Lanka's three-man pace attack of Dhammika Prasad, Nuwan Pradeep and Shaminda Eranga rarely troubled Robson and Gary Ballance.
The two newcomers put on 142 for the second wicket although Ballance had an escape when he was given out lbw soon after lunch off the bowling of Rangana Herath.
Umpire Billy Bowden's decision, however, was overturned after a video review.
Sri Lanka's frustration started to show when Pradeep bowled one over containing four no-balls.
Captain Angelo Mathews then broke through when he had Ballance caught behind for 74 by wicketkeeper Dinesh Chandimal.
Earlier, England lost captain Alastair Cook for 17 in the fourth over of the day when he edged Prasad to Kumar Sangakkara at first slip.
(Editing by Tony Jimenez)
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