Pakistan were 89-1 when bad light stopped play early and stumps were called on day four in Galle with left-handed opener Imam-ul-Haq on 46 and skipper Babar Azam on 26.
The arrival of rain ended any chances of play, adding to Pakistan's hopes of a draw to clinch the two-match series, which they lead 1-0.
But the tourists will need to make an unprecedented chase -- with another 419 runs needed to win -- if they are to make it 2-0.
The West Indies hold the record for the highest successful Test chase with 418 against Australia in 2003 at Antigua.
Sri Lanka, who declared their second innings on 360-8 in the second session after resuming on 176-5, need nine wickets on the final day to end the series level.
Spinner Prabath Jayasuriya sent back first-match hero Abdullah Shafique for 16 with debutant Dunith Wellalage taking a good running catch at mid-off before tea.
Imam and Azam played with positive intent and took their chances against the spinners to get quick runs in the weather-hit final session of play.
The pair hit six boundaries between them and Azam also smashed Jayasuriya's left-arm spin for a six over the mid-wicket region.
Earlier, De Silva made 109 before being run out at the non-striker's end and Sri Lanka skipper Dimuth Karunaratne called his batsmen back to the pavilion.
De Silva put on a key stand of 126 with overnight partner Karunaratne, a left-handed batsman who battled back pain during his knock of 61.
Beginning the day on 27, De Silva reached his ninth Test hundred with a cracking boundary off Mohammad Nawaz and raised his bat to the applauding dressing room.
Karunaratne fell in the first session after he escaped two close calls, which were unsuccessfully reviewed by Pakistan, off Nauman Ali's left-arm spin.
Nauman finally got Karunaratne, caught at short leg, where Shafique took a sharp reflex catch.
Karunaratne, who had back spasms and did not field in the Pakistan innings, got help from the physio during his two hours and 45 minutes at the crease as he went past 6,000 Test runs.
He did not take to the field in the final innings with De Silva again standing in as skipper.
Pakistan fell behind Sri Lanka by 147 runs after being bowled out for 231 in response to the hosts' first-innings total of 378.
Pakistan's highest chase in Sri Lanka came in 2015, when they made a target of 377 in Pallekele.
The tourists chased down a Galle record of 342 in the opener of the current series.
Sri Lanka v Pakistan second Test scoreboard
Scoreboard at stumps on day four of the second Test between Pakistan and Sri Lanka in Galle on Wednesday:
Sri Lanka 1st innings 378 (D. Chandimal 80, O. Fernando 50; N. Shah 3-58, Y. Shah 3-83)
Pakistan 1st innings 231 (A. Salman 61; R. Mendis 5-47)
Sri Lanka 2nd innings (overnight 176-5, D. Karunaratne 27, D. de Silva 30)
N. Dickwella c Rizwan b Naseem 15
O. Fernando lbw b Yasir 19
K. Mendis lbw b Nawaz 15
A. Mathews c Azam b Salman 35
D. Chandimal c Rizwan b Naseem 21
D. Karunaratne c Shafique b Nauman 61
D. de Silva run out (Yasir) 109
D. Wellalage c Rizwan b Nawaz 18
R. Mendis not out 45
Extras (b9, lb7, w1, nb5) 22
Total (8 wickets dec, 91.5 overs) 360
Yet to bat: P. Jayasuriya, A. Fernando
Fall of wickets: 1-27 (Dickwella), 2-47 (Oshada), 3-59 (K. Mendis), 4-100 (Mathews), 5-117 (Chandimal), 6-243 (Karunaratne), 7-278 (Wellalage), 8-360 (De Silva)
Bowling: Hasan 10-0-44-0 (w1, nb1), Naseem 12.5-1-44-2 (nb3), Yasir 21-2-80-1, Nawaz 21-4-75-2, Nauman 14-0-54-1, Salman 12-2-41-1 (nb1), Azam 1-0-6-0
Pakistan 2nd innings (target 508)
A. Shafique c Wellalage b Jayasuriya 16
Imam-ul-Haq not out 46
B. Azam not out 26
Extras (nb1) 1
Total (1 wicket, 28 overs) 89
Yet to bat: F. Alam, M. Rizwan, A. Salman, M. Nawaz, N. Ali, Y. Shah, H. Ali, N. Shah
Fall of wickets: 1-42 (Shafique)
Bowling: Asitha 3-1-6-0 (nb1), Jayasuriya 13-1-46-1, R. Mendis 9-0-31-0, De Silva 2-0-4-0, Wellalage 1-0-2-0