Teravainen cut it to 3-1 at 6:33 of the third period, scoring in front on a pass from Sebastian Aho.
Copp scored at 11:10 for the 4-1 final. It was his fifth goal of the Stanley Cup Playoffs.
He was acquired in a trade with the Winnipeg Jets on March 21.
"I think they pushed hard and we were maybe a little bit on our toes in the third but getting that fourth goal to kind of alleviate a little bit of that pressure was huge for us," Copp said. "But I think the never-die mentality has been in this team from Day One, and I've kind of felt that from walking in a couple of months ago. I feel like the new guys have kind of fit right in with that mentality."
The Hurricanes have scored six goals in the series, one of which was an empty-net goal. They scored two goals in Games 3 and 4 after ranking ninth in the regular season in goals per game (3.38).
"We obviously wanted to come here and win some games," Carolina captain Jordan Staal said. "Disappointing effort obviously, especially today but we'll have to regroup tomorrow, figure out what we're doing wrong and have a better effort and go win one at home."
NOTES: Aho, who had an assist on Teravainen's goal, became the Hurricanes/Hartford Whalers all-time playoff scoring leader with 44 points, passing Eric Staal. His 26 playoff assists are also first, one ahead of Ron Francis. … The Rangers won their fifth straight home game in the playoffs, tying their team record, done four times previously (last in 1994). … The Hurricanes are the third team in NHL history to lose their first five road games in a postseason, joining the 1992 Montreal Canadiens and the 1976 Toronto Maple Leafs. … It was Lindgren's first NHL multipoint game in the playoffs.