Horton had just passed the puck and was entering the Canucks' zone when Rome levelled him with an open-ice hit just outside the Vancouver blue-line at 5:07 of the first period.
He was on the ice for several minutes before medical staff transferred him to a stretcher.
The Bruins said Horton has been transported to Massachusetts General Hospital and is moving all extremities.
The Boston crowd started chanting: "Nathan Horton! Nathan Horton!" as he was loaded on to the stretcher.
Rome was assessed a five-minute major — served by Raffi Torres — and a game misconduct.
"That's a late, late, late hit," hockey personality Don Cherry said during the first intermission on CBC's Coach's Corner. "That's not good when its so late. (Horton) never dreamed it would be so late."
Boston coach Claude Julien told the Versus television broadcast that he had no other information on Horton, who is in the playoffs for the first time in his NHL career.
"It's a tough thing to swallow right now," Julien said. "Our spirits are OK. I heard guys saying, 'Let's do it for Horty.' So we're good to go."
After the play, several NHLers took to Twitter to comment on the hit.
Max Pacioretty, a forward with the Montreal Canadiens who was severely injured by a body check from the Bruins' Zdeno Chara during the regular season, posted: "My thoughts and prayers are with Nathan Horton and his family right now.. hoping for the best for him."
Joffrey Lupul of the Toronto Maple Leafs tweeted: "Hope Horton is OK. Scary stuff."