Crosby, who was responsible for three of the misses (36, 51 and 40 yards), atoned for his earlier errors as he finally split the uprights inside the final two minutes of extra time.
The chaotic finish commenced when Crosby missed wide left on his first effort with two minutes and 14 seconds left in the contest. Bengals kicker Evan McPherson then doinked a 57-yard try off the right post before Crosby again fired wide as time expired, sending the game into extra time.
Overtime started in similarly bizarre fashion, with Cincinnati quarterback Joe Burrow throwing a terrible interception on the very first play from scrimmage. It set up yet another Crosby kick, but once more he sailed it wide left.
McPherson would get another go at picking up the win for the home side, and he even celebrated doing so when believing he had been successful with his next 49-yard try, but that too he just crept on the wrong side of the left stick.
But, finally, Crosby nailed the fourth of his attempts to see Green Bay (4-1) come away with the road win over Cincinnati (3-2).
Before the overtime kicking drama commenced, it was Rodgers and Burrow who were trading blows in an impressive outing from both quarterbacks - although both were a bit slow to get going.
Rodgers threw an early interception, and though the Bengals didn't score any points off that turnover, Burrow would get them on the board soon after with a four-yard touchdown toss to Samaje Perine.
Green Bay would answer back on the very next series - Rodgers to A.J. Dillon for a 12-yard score - but Crosby would nod towards his kicking struggled to come by missing the extra point attempt.
Crosby would, however, prove successful with his first three field goal tries of the game - the third of which giving Green Bay a 22-14 advantage with just over 11 minutes left in the game.
The Packers had gone into half time with a slender 16-14 advantage after a thrilling finish to the first half, in which Rodgers found favourite target Davante Adams for a five-yard TD, only for Burrow to answer back almost immediately with a 70-yard scoring strike to Ja'Marr Chase.
After Crosby helped stretch Green Bay's advantage to eight in a more sedate opening to the second half, Cincinnati tied things up with three-and-a-half minutes left after an eight-yard Joe Mixon touchdown run and then a successful two-point conversion.
Cue the kicking drama that would send the game deep into overtime before Crosby finally clinched victory for the visitors.
The Packers have a divisional clash, heading on the road to face the Chicago Bears next Sunday. The Bengals, meanwhile, go head to head with another NFC North opponent as the travel to Detroit to take on the winless Lions.
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