An inferior net run rate meant Pakistan had to win big against Sri Lanka in order to pip New Zealand and join group leaders England in the semi-finals.
They racked up 212-8 after Sahibzada Farhan (100) and Fakhar Zaman (84) combined in a T20 World Cup record 176-run opening stand.
Pakistan had to restrict their opponents to 147 or less to make the last four but Sri Lanka finished on 207-6 nearly snatching a victory.
Pavan Rathnayake made 58 and Sri Lanka captain Dasun Shanaka smoked an unbeaten 76 off 31 balls.
Defending champions India will play West Indies in a virtual quarter-final in Kolkata on Sunday with the winners claiming the last semi-final spot.
"It could have been a different story had we won the toss," Pakistan captain Salman Agha said.
"Dew was a factor, and we couldn't execute our plans."
Put in to bat in a do-or-die match at the Pallekele International Cricket Stadium, Pakistan could not have asked for a better start.
Farhan led their charge with scintillating power-hitting as he and Zaman denied Sri Lanka a breakthrough until the 16th over.
Sri Lanka's sloppy fielding did not help their cause either.
Dunith Wellalage dropped Farhan on 75, while Janith Liyanage caught the batter on 76 but stepped onto the boundary cushion in the process.
Farhan breaks Kohli's record
Farhan, who also bettered Indian Virat Kohli's record of most runs in a T20 World Cup with his tally of 383, needed 59 balls to become the first batter to score two hundreds in the same edition of the tournament.
The partnership came to an end when Zaman dragged a Dushmantha Chameera delivery onto his stumps, which triggered a late collapse.
Farhan was caught in the covers off the bowling of Dilshan Madushanka after a stellar knock studded with five sixes and nine fours.
Having already been eliminated, Sri Lanka faltered early in their chase and were 33-2 having lost both openers inside five overs.
Charith Asalanka fell after a brisk 25 and spinner Abrar Ahmed (3-23) dismissed Kamindu Mendis in his next over.
Rathnayake's belligerent 58 could not force a win for Sri Lanka but knocked Pakistan out of the tournament.
Shanaka, whose job is reportedly at risk after leading Sri Lanka's poor campaign, injected fresh excitement into the dead rubber by hitting eight sixes.
Needing 28 from the last over from Shaheen Afridi, Shanaka began with a four before clobbering three sixes in a row.
Afridi recovered from the assault to finish with two fairly wide deliveries which Shanaka could not connect with.
"It was a close game," Shanaka said. "I could have finished it but well bowled to Shaheen."