Australia made 7-117 led by Mike Hussey's unbeaten 54 in reply to Pakistan's 6-149.
India's subsequent one-run win over South Africa meant both sides missed the semi-finals.
George Bailey's men finished on top of Super Eights Group Two with a 2-1 record and a superior net run-rate to Pakistan (2-1) and India (2-1).
Australia will play West Indies on Friday (2330 AEST) while Sri Lanka host Pakistan on Thursday.
All seven of Australia's wickets fell to Pakistan's spinners who bowled 18 overs in the 20-over innings.
"Call it embarrassing if you want," Bailey told his post-match press conference when asked about the nature of his side's defeat.
The skipper admitted his side had been outplayed in all three departments.
"If you look at how all the results have gone through the tournament most teams have had a result like that at some stage," he said.
"The advantage of the way we played previously (scoring two big victories earlier in the Super Eights) is that we had that buffer."
Bailey said Australia's attempt to counter top-class spinners in favourable Colombo conditions was a work in progress.
"I don't think it's any great secret that we have some deficiencies at times against spin and that showed tonight," he said.
Mohammad Hafeez (2-22) and Raza Hasan (2-14) opened the bowling and immediately put a choke hold on Australia's scoring.
Superstar allrounder Shane Watson was lbw to Hasan for eight, failing to reach 40 for the first time in five hits in the tournament.
David Warner repeated Watson's manner of dismissal, missing an attempted sweep and falling lbw for eight to Hafeez.
Saeed Ajmal (3-17) removed Bailey lbw for 15 while Cameron White (12), Glenn Maxwell (four), Matthew Wade (13) and Pat Cummins (0) also fell cheaply.
Bailey said the question of whether Australia relied too heavily on Watson (242 runs at 60.50) was a hard one to answer.
"I don't know. If he fails in the next game and we lose then you can say yes," Bailey said.
"No more so than any other team is (reliant) on their best players."
Watson has been man of the match four times in five outings in the tournament.
Pace bowler Mitchell Starc claimed 3-20 in Pakistan's 6-149.
