Smith, batting at No 6 due to the absence of injured skipper Ben Stokes, registered his third half-century in four Test appearances.
The 24-year-old wicketkeeper showcased his ability to support established batsmen and maintain the scoring rate while batting with the tail.
He shared partnerships of 62 with Harry Brook (56) and 52 with Chris Woakes (25) before both were dismissed by left-arm spinner Prabath Jayasuriya, who returned figures of 2-58 in 21 overs.
England found themselves in trouble at 67-3 in the 15th over after the morning session was washed out, with paceman Asitha Fernando claiming 3-68 in 14 overs.
Sri Lanka had experienced a similar collapse on Wednesday, losing their first three wickets for no runs in 10 balls before being bowled out for 236.
Captain Dhananjaya de Silva top-scored with 74, while debutant Milan Rathnayake set a new record for the highest score by a No 9 batsman on debut with his 72.
Dan Lawrence, recalled to open in place of the injured Zak Crawley, successfully reviewed an lbw decision against him off Asitha Fernando when on 10.
However, two balls later, the paceman trapped Ben Duckett lbw for 18, with Sri Lanka overturning the original not out decision.
– POPE BOWLED –
Asitha Fernando’s excellent bowling performance continued as he dismissed Ollie Pope, who was captaining England for the first time due to Stokes’ torn hamstring, for just six runs with a ball that nipped back through the gap between bat and pad.
Despite the setback, Lawrence appeared to be in good form until he edged a ball angled across him from left-arm seamer Vishwa Fernando to the diving wicketkeeper Dinesh Chandimal when he was on 30.
Pope’s dismissal brought Root to the crease, and the former England captain boasted an impressive record against Sri Lanka, having scored more than 1,000 runs in 11 previous Tests at an average of nearly 60, including four centuries.
However, Asitha Fernando ended a promising partnership of 58 when Root, who had scored a typically elegant 42, inside-edged an intended drive, and Chandimal dived forward to take a fine low catch.
Smith, the new batsman, confidently advanced down the pitch to drive Jayasuriya for six, and Brook completed a fifty off just 59 balls.
On the same ground where Shane Warne bowled Mike Gatting with the ‘Ball of the Century’ in 1993, Jayasuriya made a bid for the modern-day equivalent by dismissing Brook in stunning fashion. The 25-year-old rising star had added just three runs to his tea score of 53 not out when he was turned inside out by a ball from left-armer Jayasuriya that dipped in on the line of middle and leg stumps before spinning sharply and bouncing to clip the top of off stump.
Brook’s dismissal left England at 187-5, and Woakes fell in a similar manner when he was clean bowled by a quicker Jayasuriya delivery that angled in and spun significantly away from the right-hander.