Eaton took a pass from Milan Jurcina and put the puck into an open right side for his 24th NHL goal – his first since Nov. 28, 2009.
Matt Martin scored the tying goal in the third and Evgeni Nabokov made 23 saves for New York, which is on a 5-1-1 run.
Zack Smith scored a short-handed goal late in the second for Ottawa, which has lost five in a row, the past four in regulation.
Ottawa’s Craig Anderson stopped 30 shots in his 16th start in a row.
John Tavares, chosen the NHL’s first star of the month on Wednesday, failed to record a point for the second time in four games following a 12-game points streak.
PANTHERS 2, JETS 1
In Sunrise, Fla., Sean Bergenheim scored a go-ahead goal in the second period and Florida beat Winnipeg.
Kris Versteeg also scored a goal for Florida, while Scott Clemmensen stopped 20 shots.
Bryan Little scored for Winnipeg and Chris Mason made 28 saves.
Winnipeg lost for the ninth time in a row when playing the second of consecutive games. The Jets are have been shut out twice and outscored 36-7 in those.
Versteeg now has six goals and two assists in the past three games against Winnipeg.
BLUES 1, KINGS 0
In St. Louis, Jaroslav Halak stopped 21 shots and Jamie Langenbrunner scored in the second period to lead St. Louis to a win over Los Angeles.
Halak recorded his fifth shutout of the season and 21st of his career. He improved to 15-8-5 and has started six of the past seven games. Halak has four shutouts in his past seven starts.
St. Louis improved to a league-best 22-3-4 at home and has not lost a regulation game at home since dropping a 5-2 decision to Chicago on Dec. 3.
The Blues moved into a tie with Vancouver for second in the Western Conference.