Smith hit 111 before England, replying to Sri Lanka's 236 all out, were dismissed for 358 just before lunch on Friday's third day.
The Sri Lankans then lost two wickets in the final three overs of the morning session and were 2-10 at lunch, still 112 runs adrift of making England bat again with the prospect of a three-day defeat looming for the visitors.
Nishan Madushka was bowled by Chris Woakes for a duck off the third ball of Sri Lanka's innings, before Kusal Mendis also departed without scoring after nicking Gus Atkinson behind to Smith.
It completed a fine start to the day for the 24-year-old Smith, who was chosen ahead of Ben Foakes and Jonny Bairstow as England's wicketkeeper for this international summer and hasn't let his country down.
Solid with the gloves behind the stumps, Smith has also been impressive with the bat, having scored 70 and 95 in England's 3-0 Test series win over the West Indies last month.
His first Test century, in his fourth match, contained eight fours and one six as Smith guided the tail through to a big total considering England were 3-67 at one stage.
Atkinson (20) and Smith resumed with England on 6-259 overnight, and the pair shared a 66-run partnership for the seventh wicket.
Matthew Potts (17) and Mark Wood (22) weighed in with important contributions. Asitha Fernando finished with team-best bowling figures of 4-103.
Dimuth Karunaratne (4no) and Angelo Mathews (6no) were in the middle for Sri Lanka at the end of a chastening session for the tourists.
