Mohammad Hafeez (two) was the first to fall, edging Prasad to third slip, while Ahmed Shehzad (nine) was out leg before when the right-arm fast bowler returned for his second over.
Left-arm spinner Rangana Herath trapped Azhar Ali, who hit a double hundred in his last test against Bangladesh, leg before for eight to compound Pakistan's problems.
The experienced pair of Younus Khan (47) and captain Misbah-ul-Haq (20) lent some stability to the innings with a 51-run stand for the fourth wicket but Pakistan were back under pressure bother after losing both in quick succession.
Earlier, Silva, unbeaten on 80 overnight, failed to carry his bat through the innings and was the ninth wicket to fall for Sri Lanka, who resumed on 178 for three, after compiling a patient 125.
After four sessions were lost on the first two days, play started on time on Friday and it was not long before paceman Wahab Riaz struck the first blow for Pakistan.
Sri Lanka captain Angelo Mathews (19) played onto his stumps after the left-arm fast bowler had changed the angle of attack by coming around the wicket.
A patient Perera continued to thwart the Pakistan bowlers and added 37 for the fifth wicket with Dinesh Chandimal before the latter was bowled out to an arm ball from left-arm spinner Zulfiqar Babar.
The 29-year-old Silva was bogged down in the 90s for a while but finally managed to get to triple figures in 261 balls with a sublime cover drive off Babar.
Left-hander Kithuruwan Vithanage (18) gave part-time off-spinner Mohammad Hafeez his second wicket of the innings when he spooned a tame return catch back to the bowler. Silva and Vithanage added a handy 35 for the sixth wicket.
The diminutive right-hander became the third victim of Babar when he was out caught by wicketkeeper Sarfraz Ahmed while attempting a sweep shot. Riaz also finished with three scalps.
(Writing by Sudipto Ganguly in Mumbai, editing by Mitch Phillips)
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