Smith's 111 helped England to 358 all out on Friday and a first-innings lead of 122 and it looked like the Sri Lankans might be headed for a big defeat after slumping to 1-2 when Nishan Madushka and Kusal Mendis were then dismissed for ducks before lunch.
But Angelo Mathews dug in to make 65 and Kamindu Mendis was 56 not out at stumps as Sri Lanka reached 6-204, leading by 82.
Dinesh Chandimal was there at the close on 20, having returned to the middle following a trip to the hospital after being rapped on the right thumb off a rapid delivery by Mark Wood.
Sri Lanka's Dinesh Chandimal went to hospital after being hit on the thumb by a Mark Wood delivery. (AP PHOTO)
One more wicket would put the English into Sri Lanka's tail in Manchester and they are big favourites to take the lead in the three-match series.
England also look to have a gem in 24-year-old Smith, who was chosen ahead of Ben Foakes and Jonny Bairstow to keep wicket for this international summer and hasn't let his country down, impressing both behind the stumps and with the bat.
He struck knocks of 70 and 95 in England's 3-0 Test series win over the West Indies last month and now has a century in just his fourth match.
"Missing out last time against the West Indies, it was a little bit more special to get over the line," Smith said of his ton. "It's something you always dream of and wish for growing up. It is a nice one to tick off."
His century contained eight fours and one six as he guided the tail through to a strong total, considering England were 3-67 at one point on Thursday.
Gus Atkinson (20), Matthew Potts (17) and Wood (22) weighed in with important contributions in the lower order, while Asitha Fernando finished with team-best bowling figures of 4-103.
Sri Lanka had slumped to 3-6 in their first innings and didn't begin their second knock any better following the departures of Madushka and Mendis in the three overs before lunch.
After lunch, Chandimal looked in pain after his thumb was snapped back by Wood's sharp delivery and, following treatment, he went to the hospital for an X-ray. It revealed no fracture and he would eventually return.
Mathews played beautifully as he often does in Tests with England, opponents against whom he now has 10 scores of 50 or more in 23 innings.
He put on 78 with Kamindu Mendis for the fifth wicket before looping a leading edge to midwicket off Chris Woakes.
In the final hour, Wood pulled up with an unspecified injury during his run-up and part-time spinner Joe Root completed the final four deliveries of the over, taking a wicket with one of them as Milan Rathnayake was tempted into a big drive and holed out to Ben Duckett in the deep for 10.
"Both sides have toiled on a good surface," Smith said, "but to have them six down there ... you can't ask for much more than that heading into tomorrow."
