Nico Hischier made a great backhand pass to Palmieri who was streaking up the middle with 3:23 to go in the first. Palmieri got himself alone in front but was denied on his backhand by Hogberg.
Mirco Mueller took two minutes for slashing late in the opening frame and the Senators took advantage of their second power play of the period.
With 1:18 to go in the first, Ennis picked up a rebound in front after Blackwood made a tough save on a Tkachuk redirection and he made no mistake, rifling it past the Devils netminder.
Shots on goal after one were 13-12 for the Devils.
Hughes had a great chance at point-blank range three minutes into the second but was robbed on a glove save by Hogberg.
After tempers flared up in the first period, it finally boiled over into a fight as Subban and Tkachuk dropped the gloves for a spirited bout four minutes into the period.
With 15:32 remaining in the second, Kevin Rooney picked up the puck behind the Senators net and wrapped it in to put the Devils back in front.
After two, shots on goal were 16-12 for the Devils; 29-24 New Jersey overall through 40 minutes.
Devils had two power plays in the third but the most costly was a penalty to Brady Tkachuk where twice on the same penalty kill, 35 seconds apart, the Senators scored to take the lead.
Vlad Namestnikov scored on a two-on-one to tie the game at 2-2 and then on the very next shift, Connor Brown fed the puck to Chris Tierney who backhanded it past Blackwood to put the home side in front.
"It's something we're going to address," said Nasreddine. "They were bad breaks by us."
The Devils pressed for the equalizer for most of the period, ultimately peppering 20 shots on Hogberg. It eventually paid off when Palmieri tied it with 3:21 to go. Palmieri swooped in to pick up great one-handed pass from Jesper Bratt at the top of the crease and put it into the open net to knot the game up at 3-3.
"We just didn't get down or negative. We knew we had the guys in here with the skills. We had to be positive," said Severson. "It's good to see us getting pucks on net and getting pucks through. If we continue that trend maybe we won't run into such hot goalies on the schedule."
At the end of regulation, shots were 48-35 for the Devils (19-11 in the third).
Overtime was a back-and-forth affair with some great chances on both sides.
Severson was denied on a breakaway by Hogberg in the opening minute of the extra frame. Just over a minute later, Blackwood made a brilliant save on a wraparound by Batherson and moments later, Hogberg denied Bratt on a breakaway as action broke from end to end. In the final minute of the game, Hogberg again made a great stop, this time on Will Butcher.
"We had such good looks and he made such great saves. We did a good job forechecking, coming through the neutral zone. Our goalie came up huge as well," Severson added.