Filip Gustavsson made 26 saves for the Wild (41-28-6), who have lost three of their past four (1-2-1).
Minnesota is one point ahead of the St. Louis Blues for the first wild card into the Stanley Cup Playoffs from the Western Conference.
"I thought it was a really hard-fought game," Wild coach John Hynes said. "Both teams played with playoff intensity. We really battled, and I thought we carried play and stuck with it. Fortunately, we were able to rebound and get the tying goal and at least get it to overtime (for a point)."
Hughes gave the Devils a 1-0 lead at 12:30 of the first period on a wrist shot from the left face-off circle.
"We got back to back wins against a really good team, so definitely helps our confidence with the way we can play," Hughes said. "So, we just got to keep going as a team and hopefully clinch a playoff berth soon."
Vinnie Hinostroza tied it 1-1 when he deflected a point shot from Jake Middleton in the slot at 2:57 of the third period.
Gustavsson kept the game tied 30 seconds later when he was able to trap a puck that popped into the air and onto his backside after a shot from the top of the right circle by Brett Pesce. A video review confirmed the puck did not cross the goal line as it sat on Gustavsson's back.
Nico Hischier scored on a redirection at the left post at 9:12 of the third period to give the Devils a 2-1 lead. It was Hischier's fourth goal in the past two games.
"I went backdoor and was expecting or just trying to be there and be ready when the puck was coming," Hischier said. "[Stefan Noesen] got it there."