Fisher made birdies on three of his first six holes and then birdied four of the first five holes on his back nine.
Gallacher shot a bogey-free round.
The co-leaders were followed by George Murray of Scotland at 5 under.
Martin Kaymer, who holed the putt that kept the Ryder Cup with Europe less than two weeks ago, shot a 69 on the Oceanico Victoria Golf Course.
Ryder Cup teammate Francisco Molinari finished at par, while Europe captain Jose Maria Olazabal was 4 over.