With the series tied 2-2 going into the match, the result was pivotal, with the winning side having the chance to close things out two nights from now.
The Ducks took the lead in the first period on the power play when Beckett Sennecke managed to finish the shot right in front of goal.
It was Sennecke's fourth straight game finding the back of the net.
The Golden Knights did not take too long to answer, as they also scored their opener on a man advantage. Pavel Dorofeyev stole the puck from his opponent before skating in and firing his shot past Lukas Dostal.
While the second period did not have a goal, the Ducks dominated proceedings, outshooting Vegas 17-6. Yet, they could not find the go-ahead goal to take the lead.
Despite being second best for much of the match, the Golden Knights went in front less than five minutes into the third. Tomas Hertl was the one to get his stick on the loose puck in front of goal. He converted from close range to put his team ahead for the first time in the night.
Anaheim were now the ones pushing for the equaliser, which came for them with 3:03 left in regulation. Olen Zellweger took his time after receiving the puck from Cutler Gauthier. Keeping a cool head, the 22-year-old shot it past Carter Hart to make it 2-2.
Nothing could separate the two teams in regulation, meaning overtime would be required to determine a winner.
It was Vegas who found the all-important fifth goal at 4:10 of overtime, as Dorofeyev scored his second of the evening via a rebound from Jack Eichel's shot, to seal victory for his side.
Game 6 of the second-round series takes place on Thursday in Anaheim.