The Swans conceded the first 32 points but finished with a flurry of nine goals to one in the final quarter to post their fifth win of the season.
After a wayward beginning, superstar Lance Franklin (three goals) got his side ahead for the first time with about 10 minutes left on the clock.
Ben Ronke (three goals) and Oliver Florent were among those to get in on the action as Sydney ran out convincing 16.13 (109) to 10.8 (68) winners.
The win was soured by a concussion to key-position player Paddy McCartin, who was subbed out of the game in the third quarter having copped a knee to the head.
McCartin, who was delisted by former club St Kilda in 2019 after suffering a string of concussions, cut a despondent figure in the sheds after failing a concussion test.
Sydney coach John Longmire said the 26-year-old was "feeling good" and wanted to get back on the field.
"The (doctors) are ultra conservative with him as you'd understand," he said.
"He was disappointed he couldn't get back on the ground."
The Swans ended the game with three on the bench after forward Logan McDonald also failed a concussion test.
Callum Mills was heavily involved for Sydney, topping the match possession count with 37 to go with five tackles, three score assists and a goal.
"We'll look at why we took a bit of time to get out of the blocks," Longmire said.
"It was a ten-day break, we just looked like we needed to get going a bit sharper a bit earlier."
As they did in their upset win against Geelong last round, Hawthorn started on fire, slotting the first five goals before the Swans even got on the board.
The tension was noticeable on field for Sydney, with Dane Rampe having a verbal crack at teammate Chad Warner after a defensive lapse resulted in a Hawthorn major.
After the sluggish start, Sydney rebounded with the last three goals of the opening term to reduce the margin to 14 points at the first break.
The second quarter was a scrappy affair - the Swans errant with the boot, with Franklin at one stage sitting on an uncharacteristic 1.4 in front of goal.
Sydney came to life towards the end of the third, going back-to-back to help reduce the Hawks' lead to 62-53 leading into the last quarter.
Jacob Koschitzke kicked three goals for Hawthorn, who are 3-3 after six rounds.
"They're one of the benchmark teams," Hawks coach Sam Mitchell said.
"We learned a little bit about ourselves and what we need to do. When we produce our best, we match it with a side like that."
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