Eichel, who was acquired Nov. 4 in a trade with the Buffalo Sabres and recently made his Vegas debut after having artificial disc replacement surgery, has three goals as a Golden Knight, including two game-winners.
"That's what you miss, just making a difference," Eichel said. "You're just trying to help the team win. As a competitor, you miss competing and being out there and having moments like that ... helping win hockey games."
Jonathan Marchessault also scored for the Golden Knights. Robin Lehner made 39 saves and earned his 150th career win.
Brady Tkachuk scored the lone goal for Ottawa, and Anton Forsberg made 40 saves.
Vegas entered the game with the worst power play in the NHL since Jan. 24 at a bleak 5.4% (2 for 37) conversion rate and was already 0 for 1 in the game.
Ottawa's Thomas Chabot was called for tripping with 54 seconds left, giving the Golden Knights their second power-play opportunity.
"I don't know if that's a penalty or not, I haven't looked at it," Ottawa coach D.J. Smith said after the game. "That's a tough way to lose. If we play like that, we'll win our share. That's how we have to play."
Eichel, Vegas' newest $10 million dollar man, sent the crowd into a frenzy by scoring the game-winner after the puck caromed to him on the left side and he fired it past Forsberg.
"There's a handful of guys that can corral a puck, get it off that quickly and stick it in the back of the net in that spot in the world — he's one of them," Vegas coach Peter DeBoer said. "That's exactly why you go get him."
While Eichel was credited with the winner, Lehner may have had the play of the night early in the third period. Nick Paul strolled in from the corner and fed Zach Sanford in front of the crease, but Lehner stopped the chance with a spectacular glove save.
"He had a lot of big saves, obviously that save in the third was key," said Eichel, who was also teammates with Lehner in Buffalo. "We've gotten really good goaltending and some key saves at key times and it's given us a chance to win hockey games. A lot of credit to Lenny tonight."
It was clear from the start the game was headed for a goaltenders' duel after a scoreless, albeit lively, first period in which the Golden Knights outshot Ottawa 17-15 and held an edge in scoring chances (13-8) and high-danger chances (4-3).
"I think when you look around the league, if one goalie at one end is outplaying the goalie at the other end or even, you've got a chance to win," DeBoer said. "And if he's not, you're gonna be in tough this time of year. I thought their goaltender was excellent, and I thought Lenny went toe-to-toe with him and was great and that's what we need."
Both teams finally got on the board in the second.
Marchessault raced in untouched from Vegas' zone and through the slot and sent a wrist shot high over Forsberg's glove. Marchessault extended his point streak to five games with four goals and two assists in the span.
The lead didn't last. Ottawa's power play, which had struggled on the road, scored the equalizer after Vegas' Michael Amadio was called for tripping. The Senators methodically worked the puck to the goal line, where Colin White was in the right spot to feed Tkachuk for a tap past Lehner.
The Senators came into the game with the second-worst road power play in the NHL, having converted just 13.2 per cent (10 of 76) of their opportunities before their first chance Sunday.
NOTES: Vegas D Brayden McNabb was placed on injured reserve with an upper-body injury. ... Ottawa is now 0-4-0 in March and has lost five in a row overall. ... The Senators dropped to 9-15-3 on the road this season. ... Vegas F Reilly Smith has at least one point in four of his last five games, with two goals and three assists. ... Vegas is now 7-1-0 all-time against Ottawa and has won the last six meetings. ... Marchessault leads all skaters with 11 points (6 goals, 5 assists) in the eight meetings with Ottawa
Senators: Conclude a five-game road in St. Louis on Tuesday.
Golden Knights: Open a five-game trip in Philadelphia on Tuesday.