Head hammered 101 off 72 balls and propelled Australia to impressive 313-7 after they were left with only 13 fit players hours before the start of the three-match series.
In-form opening batter Abdullah Shafique (81) and veteran Azhar Ali (78) had carried Pakistan to 170-1 and denied Australia a breakthrough in the first session after resuming on 90-1 before the visitors took control of the series deciding test.
Sedate half-centuries by Cameron Green and Alex Carey earned them Australia's highest sixth-wicket stand in Pakistan, worth 135. But they were broken up after lunch and Australia was all out on the strike of tea.
No. 9 batter Nauman Ali successfully defended 18 balls without scoring and saw off the last eight overs with Rizwan after Pakistan had slipped to 414-7.
Pakistan vs Australia 2nd Test, Day 5: Get the Live Cricket Scores, Match Updates, Highlights, Full Scorecard, Results from the PAK vs AUS 2nd Test in Karachi.
Babar was unbeaten on 102 off 198 balls and opening batter Shafique followed his maiden test hundred in the drawn first test with an undefeated 71 off 226 balls to take Pakistan to 192-2 at stumps.
Pakistan vs Australia 2nd Test, Day 4: Get the Live Cricket Scores, Match Updates, Highlights, Full Scorecard, Results from the PAK vs AUS 2nd Test in Karachi.
Pakistan, replying to Australia's huge first-innings total of 556-9 declared, was bowled out for just 148 after having spent more than two days toiling in the field.
Carey and Mitchell Starc, who was unbeaten on 28, nearly played out the entire last session as they put on 98 runs on a slow wicket to the frustration of Pakistan’s bowlers.
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Khawaja and Steve Smith (72) shared a 159-run third-wicket stand before Pakistan broke it with the second new ball when Faheem Ashraf took a brilliant one-handed low catch in the slips to dismiss Smith off Hasan Ali (1-31).
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Swepson first toured with Australia's test side five years ago and has consistently been Nathan Lyon's spin deputy in the past few years, but has not been able to make the final XI.
Pakistan's first home Test against Australia in 24 years ended in a tame draw on Tuesday with the hosts posting 476/4 declared and 252 for no loss in their two innings, and the visitors responding with 449 all out -- a total of 1187 runs for the fall of just 14 wickets over five days.
Pakistan openers Imam-ul-Haq and Abdullah Shafique scored unbeaten centuries as Pakistan scored 252 without losing any wicket on the fifth and final day of the first Test match.
Pakistan vs Australia, 1st Test, Day 5: Get the Live Cricket Scores, Ball by Ball Commentary, Full Scorecard, Latest Updates, Highlights, Live News from the Pakistan vs Australia, 1st Test, Day 5, Rawalpindi Cricket Stadium, Rawalpindi.
Pakistan had declared its first innings at 476-4 late on Day 2 on the back of big centuries from Azhar Ali (185) and Imam-ul-Haq (157).
Pakistan vs Australia, 1st Test, Day 4: Get the Live Cricket Scores, Ball by Ball Commentary, Full Scorecard, Latest Updates, Highlights, Live News from the Pakistan vs Australia, 1st Test, Day 4, Rawalpindi Cricket Stadium, Rawalpindi.
Khawaja, whose family migrated to Australia when he was four, capitalized on two dropped catches to score 97 while his opening partner David Warner (68) and Marnus Labuschagne (69 not out) also scored aggressive half-centuries.
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