Cats veteran Hawkins surpassed long-time teammate Joel Selwood's club record for most games played in his 356th appearance.
But it was the Giants who were celebrating at GMHBA Stadium on Saturday night after their 11.12 (78) to 11.8 (74) victory snapped a three-match losing streak.
The result sent GWS (7-4) back into the top four above their opponents, and condemned fifth-placed Geelong (7-4) to a fourth straight defeat.
It came at a cost, with experienced midfielder Stephen Coniglio substituted out of the game during the second quarter with a shoulder injury.
Geelong trailed by 31 points during the third quarter but rallied at home in front of 30,821 fans, kicking five consecutive goals to hit the front in the final term.
The Cats dominated territory in the last quarter, winning a landslide inside-50 count (24-5), but couldn't make it count.
They were denied by Greene's brilliant right-foot snap with more than seven minutes remaining and stellar defensive efforts by the Giants, including a stunning mark by Leek Aleer in the dying stages.
Greene shook off an ankle concern after landing awkwardly from a marking contest and finished with two goals from 24 disposals.
Jake Riccardi (four goals) and Coleman Medal contender Jesse Hogan (two) were also important in attack, while Tom Green racked up 28 touches despite close attention from Mark Blicavs at stoppages.
Max Holmes (32 disposals, five clearances) and Zach Tuohy (24, six) starred for Geelong, and Hawkins (two goals) kicked multiple majors for the first time since round four.
The lead changed hands five times in the first half before GWS took control for a period, turning a 14-point quarter-time deficit into a 19-point buffer at the main break.
The visitors piled on six goals to one in the second term, their midfield dominance crucial in a flurry that came despite Coniglio being forced off.
The Giants led 8.9 to 6.2 at halftime and the margin swelled to a game-high 31 points early in the third quarter.
At that stage, the game looked over.
But Geelong gradually hauled themselves off the canvas and a pair of goals to Tyson Stengle at the start of the final term levelled the scores.
Geelong's Brad Close was denied a brilliant goal off the deck, his kick ruled touched off the boot by Harry Perryman on review, but it was enough to give the Cats the lead again.
Greene kicked the final goal of the match on the outside of his right boot under all sorts of pressure to put the Giants back in front, before they hung on in the dying stages.
