Eilers defeated Matt Murray at 2:20 in extra time, shortly after Gates goalkeeper Laurent Brosswitt saved Drake Batherson’s expense.
Before the victory, Jets had to clear two goals from Nxteen.
They forced extra time thanks to Josh Morrisey’s grid with 77 seconds on the clock as Brossuit gave way to a sixth striker.
Adam Lowry and Kyle Connor scored the other goals for the Jets (2-1-0), who beat 41-28 on penalties.
Josh Norris, Chris Tierney and Alex Galchenock managed to score the Senate goals (1-1-1).
The Senators dominated the match in the first period, beating their rivals by 18 shots to 9. They were rewarded by scoring the first two goals of the night.
Norris was the first to defeat Prosuit in a power game. He scored the first goal of his National Hockey League career while sliding against a goalkeeper sled.
Connor came close to match in tough play, but his shot went off the post. On the other end, Tierney didn’t miss his chance when he fired a shot behind Brossoit to score his second goal of the season.
Lowry allowed the Gates to close in on a late goal after Neil Bionk’s shot was pushed back behind Murray, who was starting for the third time in a row.
The second period was like the previous one. The Senators once again had the upper hand in shots on target, 14-8, while Alex Galchenyuk gave his team a two-goal priority with his first goal of the season over tough play.
After less than five minutes, Connor narrowed the Senate lead to one goal again, also in the power game.
It was the first of three matches in five days between the two clubs. They will meet again on Thursday evening in Ottawa before the Saturday night fighting in Winnipeg.
The Jets played the match without the services of forward Patrick Lane with an upper body injury for the second match in a row. Her condition is re-evaluated on a daily basis.
For the Senators, rising striker Tim Stotzel, who has suffered a minor injury, did not appear in the match. After scoring a goal in two matches, Stützle will also be re-evaluated on a daily basis.
Hughes beats Lavriniere
In a duel between the top two draft picks, Jack Hughes had the upper hand over Alexis Lafrenière by scoring his first two LHH goals in the 4-3 victory of New Jersey Devils over the New York Rangers.
After three matches in the NHL, Lafrenière is still looking for a first point.
Hughes added a pass, his fourth, past Miles Wood, the last of the demons.
Hughes got all these points in the second half.