The tourists, set a challenging target of 377 runs, moved from their overnight score of 230-2 to 321-3 in their second innings by lunch on the fifth day.
Skipper Misbah-ul Haq was unbeaten on 27, with Pakistan just 56 runs away from a win with seven wickets in hand and two sessions remaining in the match.
If Pakistan achieve the target, it will be the first time a visiting team will have scored more than 300 runs in the fourth innings to win a Test on Sri Lankan soil.
Pakistan added 25 runs to their overnight score when they lost the wicket of Shan Masood, who was stumped by Dinesh Chandimal off Tharindu Kaushal for 125.
The Kuwait-born left-hander's maiden Test century came during a third-wicket stand of 242 with Younis after Pakistan had been reduced to 13-2 on Monday afternoon.
Sri Lanka claimed the second new ball as soon as it was due in the 80th over, but the seamers were unable to make an impression on the easy-paced wicket.
Pakistan won the first Test in Galle by 10 wickets and Sri Lanka took the second in Colombo by seven wickets.
The two teams will play a limited-overs series of five one-day internationals, followed by two Twenty20 games, from Saturday.
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