Rickie Lambert put the hosts into the lead at St Mary's just after the half-hour mark when he prodded home a loose ball in the Canaries' penalty area.
The Saints were full value for their lead at that stage and looked set to go in at the break a goal to the good until Gazzaniga gifted the visitors an equaliser.
The summer signing from Gillingham failed to get down in time to deal with Robert Snodgrass' free-kick, allowing the ball to squeeze between his body and the ground.
All eyes were focused on Mark Clattenburg before the game, as the referee took charge of his first match since Chelsea alleged he used inappropriate language towards John Obi Mikel during their match against Manchester United on October 28.
The official enjoyed a fairly comfortable comeback, although both teams had complaints about the goals they conceded. He showed a yellow card to a player from each team and also rejected appeals from Norwich for a late penalty after Luke Shaw's desperate tackle on Snodgrass in the area.
The point means Chris Hughton's side have now drawn their last four away games in league action, a feat the club have not managed since the 1979/80 season when none of their line-up were actually born.
For Southampton they have at least now taken seven points out of their last nine, though they will perhaps look back on a chance missed to make it three victories in a row.
In truth, though, they never hit the same heights as they managed in their impressive triumph over Newcastle at home on Sunday. While dominant for long periods, they failed really to test Mark Bunn, standing in for the injured John Ruddy, in the visiting goal on enough occasions.
Indeed, their most meaningful attack of the first half resulted in them taking the lead, Lambert in the right place to stab home after Adam Lallana's free-kick from the left flank had struck him on the arm and then ricocheted off his former Rochdale team-mate, Grant Holt, back into his path.
Norwich pulled level on the stroke of half-time as Snodgrass' rather tame set-piece squirmed past Gazzaniga, though the 20-year-old did make make amends to some degree in the second half with a close-range save to keep out a Holt header at the far post. Southampton, too, felt the decision to award the free-kick for the goal was harsh on full-back Nathaniel Clyne.
Despite the best efforts of both teams a winner couldn't be found in the second half, Bunn diving acrobatically to his right to paw away a low, curling effort from Jason Puncheon before Shaw's last-ditch lunge stopped Snodgrass from getting a chance to score his second of the night.