Australia captain Aaron Finch was relieved his side survived a Pakistan revival at Taunton.
A gripping match saw Australia put 307 on the board before bowling Pakistan out for 266, but until the late stages the contest was firmly in the balance.
And that was reflected in Finch's verdict as he acknowledged some big hitting down the order from Pakistan had given his bowlers a headache.
Pakistan looked to be sinking at 160-6 but it was a different story at 264-7 after Hasan Ali, Wahab Riaz and captain Sarfraz Ahmed led a big-hitting rearguard action.
Mitchell Starc made a key double breakthrough at that stage though, firstly when Australia used a DRS review to show Riaz nicked a delivery through to Alex Carey, and then when bowling Mohammad Amir for a duck.
Glenn Maxwell threw down the stumps to run out Sarfraz and win the match, with Australia thankful they avoided another defeat after losing to India last time out.
That's the game.— Cricket World Cup (@cricketworldcup) June 12, 2019
Glenn Maxwell with an astonishing run-out to end proceedings. Wahab threatened to do something special for a brief period but the bowlers have done their job – it's a 41-run win for the reigning champions. #CmonAussie #CWC19 pic.twitter.com/xjGInDB7mI
Finch agreed Pakistan had put his players under pressure, saying: "They certainly did. It's always tough when you've got guys like Hasan and Wahab coming out and swinging.
"When they start to get on a roll it can be tough to stop. We just had to bowl our best ball, whether that was a length ball or whether it was a yorker. You had to commit to that ball 100 per cent. We saw that if your execution is slightly off on a small ground like this you go for six.
"We didn't bat out 50 overs, which was really disappointing. When you go in with the extra batter you stack the batters to do that job.
"We probably tried to go a little bit too hard too early and ended up probably 20-30 runs short."
Sarfraz said his bowlers came back well after disappointing opening spells allowed Australia to start well, with David Warner going on to make 107 and Finch 82.
"We conceded too many runs in the first 20 overs. Except for Amir the other bowlers did not bowl really well," Sarfraz said at the post-match presentation.
Amir took 5-30 but had little support, while Pakistan were left to lament a number of batsmen failing to convert reasonable starts into big innings.
"If we want to win, the top four have to make runs," Sarfraz said.
Pakistan face India next, and already face a battle to stay in semi-final contention.
A win over India would lift spirits and Sarfraz said: "We will try our level best."
Man of the match Warner admitted Australia should have gone on to reach 340 or 350 after their strong start.
"Credit to the way Pakistan bowled," Warner said. "Their second spells were fantastic. They hit their lines and lengths and made it hard for us. They bowled very straight lines to me and gave me no width."
Assessing the tense finish, he added: "It was probably a lot closer than we expected but I thought it was a great game."
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