Gaborik stole the puck at center ice, had his first attempt stopped and then scored on the rebound. It was the 14th hat trick of his career.
His other two goals both came in the first period and the Rangers held a 2-0 lead after the opening stanza.
Brad Marchand and Milan Lucic tallied for Boston to tie it in the second, but Tayylor Pyatt produced a goal for the Rangers to give them a 3-2 advantage heading into the third.
Nathan Horton again tied it for the Bruins with 4:23 to go in regulation.
Henrik Lundqvist stopped 26 shots for New York to help prevent the Rangers from opening 0-3 for the second straight year.
Boston had won its first two games. Tuukka Rask made 29 saves for the Bruins.