New York's Marc Staal failed in his attempt to clear the puck out of his own zone and Jason Arnott grabbed it at the right point. His pass slid to Semin in the slot, from where he put a one-time past Henrik Lundqvist.
Semin assisted on a goal by Alex Ovechkin with 6:16 to go in regulation that forced the overtime period.
The Rangers produced the first score of the game 1:56 into the third, getting a goal from Matt Gilroy. It was the first post-season game in Gilroy's career.
Michal Neuvirth made 24 saves for Washington, which came into the playoffs as the No. 1 seed in the Eastern Conference. Lundqvist stopped 31-of-33 shots.
The second game in the best-of-seven series will be hosted by Washington Friday.