The Rangers have won four straight, all on the road. Alexandar Georgiev made 26 saves.
Ottawa led 2-0 after Josh Norris’ goal early in the third period, but the Senators unravelled late. Kreider scored a power-play goal by beating Matt Murray in close with 5:23 left.
Murray, who stopped 22 shots, left the game following the goal, leaving Anton Forsberg to close things out. Forsberg gave up a goal to Lindgren on the first shot he faced with 4:08 left.
Goodrow scored the winner with 2:03 left by deflecting a long shot from Sammy Blais past Forsberg.
Nick Paul had the other goal for Ottawa, scoring on the Senators’ first shot after a setup from Tim Stutzle.
Paul was moved to center between Stutzle and Connor Brown as Ottawa will be without Shane Pinto for at least a week after he was hurt in Thursday’s game against the San Jose Sharks.
Georgiev, making his second start of the season, made an incredible point-blank save on Artem Zub in the final minutes of the second period.
NOTES: Ottawa F Scott Sabourin and D Michael Del Zotto were healthy scratches. Jarred Tinordi, Libor Hajek and Greg McKegg were healthy scratches for New York.
Rangers: Host Calgary on Monday night.
Senators: Host Washington on Monday night.