Bale delivered the perfect free-kick after 21 minutes to help Saints secure their 11th win of the season and a place in the top four.
Robert Earnshaw gave Norwich the lead with his 14th goal of the season and, after Bale's equaliser, Trinidad and Tobago striker Kenwyne Jones supplied the second-half winner.
Norwich only seriously threatened towards the end when veteran defender Dion Dublin was sent forward in a desperate search for an equaliser.
It needed two outstanding saves from Southampton goalkeeper Kelvin Davis to stop the Canaries grabbing an equaliser.
Southampton have now won five of their last six matches and collected 20 out of 27 points but Norwich came close to earning a draw.
Earnshaw scored on the turn with a low right-foot shot after Andy Hughes had headed down a cross from Adam Drury.
Bale's goal just before the break came with a free-kick from 20 yards which he bent into the empty net with the Norwich defence in disarray.
Bale had failed with two earlier free-kicks, one of them into the side-netting with goalkeeper Paul Gallacher beaten.
In the second half Gary Doherty should have scored in the 57th minute for Norwich when he was left with a free header at a corner by Drury but Southampton took control.
Grzegorz Rasiak went close with a hooked shot as Bale got down the left and Earnshaw was booked by referee Uriah Rennie for diving in the 63rd minute.
Two minutes later Southampton were in front when Jermaine Wright took a corner on the left and Jones rose above the goalkeeper to head in his seventh goal of the season.
Davis distinguished himself in the 70th minute saving from Peter Thorne with his feet following a free-kick and, with four minutes to go, the goalkeeper plunged to his right to push away a goalbound shot from Dickson Etuhu.
Southampton had Jones booked for foul on Carl Robinson after Norwich manager Peter Grant had reshuffled his squad.
Grant sent Dublin up front, brought on Craig Fleming at the back and also moved Robinson into a defensive position.
Southampton had to survive ten minutes of frantic Norwich pressure at the end and Dublin went close with a header which had Davis scrambling across his goalmouth.