Bangladesh are leading by 313 runs at the end of third day's play.
Sri Lanka so far scored 229 runs for 3 wickets from 79 overs after Bangladesh declared on 541 for 7.
Miraz, Taijul, Taskin picked up a wicket each.
Dimuth Karunaratne is unbeaten on 85 runs.
Taskin, who is playing Test cricket after 4 years, picked up the wicket of Oshada Fernando to provide a much-needed breakthrough for Bangladesh.
Taskin has been bowling well on 3rd day and he got his reward for that.
It was angling down leg, and Oshada strangled a catch down the leg side! Excellent footwork by Liton as he dived to snaffle it with both gloves.
Sri Lanka are 171 for 2 and still trailing by 370 runs.
Mehidy and Taijul both settled into an excellent rhythm and kept putting pressure on the Sri Lankan openers for the last 7-8 overs.
Miraz finally broke the 114-run opening partnership right on the stroke of tea.
Thirimanne scored 58 runs before departing while his partner Dimuth Karunaratne is unbeaten on 43.
Sri Lanka trailing by 427 runs with the last session of day 3 to be played.
Bangladesh bowlers are toiling hard to break the opening partnership of Dimuth Karunaratne and Lahiru Thirimanne.
This duo has already put up a 94-run opening partnership in 29 overs and looking comfortable in the middle.
Thirimanne bagged his 10th Test half-century meanwhile.
5 bowlers have been used so far by skipper Mominul Haque.
Taskin Ahmed was the pick of the bowler and he looked really sharp in his 7 overs.
Lahiru Thirimanne escaped an LBW shout off Taskin just before lunch.
He was given out before the decision was overturned as he reviewed.
Taskin bowled 4 overs, gave away only 4 runs and 3 of them were maiden overs. He was looking really sharp.
Sri Lanka scored 11 runs from 8 overs before lunch without losing any wicket.
Bangladesh declared on 541 for 7.
Mominul emerged from the dressing room and gestured to his batsmen to come back. Just over half an hour of batting for Sri Lanka to do before lunch.
Mushfiq was batting on 68 with Taskin being on 6.
Bangladesh batted 173 overs, meaning more than 1000 balls in their innings.
Out of the 15 sessions in the match, the Tigers batted almost 7 of them.
Shanto and Mominul bagged centuries while Tamim, Liton, Mushfiq bagged fifties.
Bangladesh have go past 500 runs in the first innings in Kandy.
Liton Das, though, departed soon after reaching his 7th Test fifty while Mushfiq is still unbeaten on 55.
Miraz later joined Mushfiq but he too went back to the pavilion after a little DRS drama in the middle.
He was given out and not out in two consecutive deliveries in the same over of Suranga Lakmal, both were overturned before being caught behind in the same fashion in the third attempt.
Bangladesh are 519/6. They started the day on 474/4 after the play was halted in the last hour yesterday.