Terry gave the Ducks a 1-0 lead at 1:46 of the first period with a one-timer from the right circle during a power play.
Vrana tied it 1-1 at 2:01 when he finished a backhand centering pass from Justin Faulk, who had two assists.
"After we get scored on, we have a good shift after that," St. Louis coach Craig Berube said. "That's really important."
Nesterenko, who was playing in his third NHL game, responded for the Ducks at 4:05, scoring on a one-timer between Faulk's legs to make it 2-1.
"Great play by Terry getting it out to [Mason] McTavish there, and then he found me for a wide-open net, so all I had to do was put it in," Nesterenko said. "Before the game, [McTavish] was like, 'I think I feel it coming.' He made a great play, so all the credit to him."
The Ducks scored on their first two shots against Binnington, who did not play in the previous four games.
"Early on, I wanted to find that rhythm of the game and feel the flow of the game," Binnington said. "The first two [shots] went in, so it took a little bit, but then we really played a very strong game. Got a lot of chances, a lot of opportunities working at both ends of the ice, so I was just happy to be back out there playing."
Blais tied it 2-2 at 15:14 of the first, and Schenn scored his 20th of the season at 15:33 to put St. Louis in front 3-2.
Schenn then scored again at 1:02 of the second period to push it to 4-2. Terry slipped and lost the puck near his own blue line, and Schenn scored off a pass from Vrana on the ensuing 3-on-1 rush.
Kapanen scored 29 seconds later to make it 5-2 when Torey Krug's point shot deflected in off him.
"Sometimes starting a game [with an early lead] can [be a] detriment because it just feels like you're going to do whatever you want," Eakins said. "That's certainly not the case against that team. They're too deep. They've got lots of guys that have been through the playoffs."
Kapanen scored his second goal 15 seconds into the third period during a 4-on-4 to extend the lead to 6-2.