Vanecek, acquired in a trade with the Washington Capitals on July 8 for a second- and third-round pick in the 2022 NHL Draft, earned his seventh NHL shutout and first since a 2-0 win at the Arizona Coyotes on April 22.
"Honestly, it was a really good game and the guys played really well," Vanecek said. "The guys were just working hard, blocking the shots and doing everything for me."
Jack Hughes scored a power-play goal at 2:59 of the third period for the Devils (5-3-0), who have won their past two games and four of five.
"We're getting comfortable playing tighter games, we're smarter with the puck," New Jersey coach Lindy Ruff said. "There were too many times last year I talked about our puck management that led to opportunities for the other team to win the game.
"We had a lot of games where we'd be winning by a goal and a penalty may have led to an opportunity for the opponent to tie the game, maybe win it in overtime or a shootout, But I think we're in a different place."