The Nets triumphed 112-85 over the New York Knicks on Wednesday after removing the interim tag from Vaughn's title, appointing him on a permanent basis.
Brooklyn have a 3-2 record under Vaughn, having started the season 2-5 under Steve Nash, who was fired last week. The Nets have also been playing without the suspended Kyrie Irving for their past four games.
"We don't want this to be a honeymoon stage for us," Durant told reporters, after recording a triple-double against the Knicks with 29 points, 12 rebounds and 12 assists.
"We want to continue to keep pushing through, keep finding ways to get better, finding ways to turn those weaknesses to strengths and keep pushing from here.
"We were struggling, we've been through a lot. Guys got pride, individual pride, and we just wanted to come out and play better. That's what we've been doing."
Durant presented Vaughn with the game ball afterwards, amid upbeat scenes from a Nets locker-room which has been through plenty this season.
"I was excited for him," Durant said. "I know the work that he puts in every day. I know how much he cares about the development of each player, and this team as a whole.
"Look forward to playing for him. All the guys have responded to how he wants us to play, so I'm looking forward to how we progress after this."
The Nets' recent change in results has coincided with keeping opposition sides below 100 points in their past four games. The run marks the first time Brooklyn have done that since the 2014-15 season.
"We've already taken a big jump last four games, teams under 100 points, which has been great," Vaughn said. "So definitely concerted effort…. We really came out with the right mindset."
On getting the job permanently, he added: "I guess I was the write-in candidate in the minds of elections right now, but I’m OK with that. I said to my wife, I might not have been her first choice and we’ve been together 20 years, so it can all work out. So, off we go."
The Nets' latest win improved their overall record to 5-7, leaving them 11th in the Eastern Conference.