Ottawa was unable to score on a 49-second two-man advantage late in the first period, but the Senators made the most of the remainder of their power play as Shannon, standing right on the lip of the crease, tipped in Jason Spezza's shot for a 1-0 lead.
Brodeur and McElhinney were at their best in the second, making a number of point-blank saves. Brodeur needed to make a big save on Spezza early in the period and a glove save on Marek Svatos later.
McElhinney looked great with a big stick save on Travis Zajac, then later he stopped Zubrus with the Devils on the power play.
"He made a couple of real timely saves," Senators coach Cory Clouston said of McElhinney's performance. "He was patient tonight and to me that was the most important thing."
The Senators took a 2-0 lead 55 seconds into the third when Erik Karlsson shot from the boards and Neil knocked the puck out of the air and past Brodeur. The play was reviewed, and officials ruled Neil's stick was below the crossbar when it hit the puck.
"I knew I didn't touch it," Neil said. "I was waiting for it to come down, but the way it was going I just needed to follow it."
The Devils appeared to cut the lead in half less than a minute later, but Colin White's apparent goal was waved off due to goaltender interference.
New Jersey finally beat McElhinney a few minutes later on a power play with Rolston's shot from just inside the blue line.
McElhinney made a huge stop on Ilya Kovalchuk late in the third to preserve the lead, before Neil added an empty-netter.