Striker Robertson, who only arrived on loan from Sheffield United last week, cut in from the left wing and curled a delightful 20-yard shot into the top corner to give his new side the lead on the half-hour mark.
The game was decided in controversial circumstances with a penalty for Southampton.
Dejan Stefanovic was ruled to have brought down Adam Lallana and the referee pointed to the spot.
Veteran defender Stefanovic protested and was shown the red card for foul and abusive language before David McGoldrick stepped up to slot home the penalty for his fifth goal of the campaign.
In a lively opening to the match, both sides hit the woodwork.
McGoldrick was the first for Southampton when his shot from eight yards cannoned off the post with the goal at his mercy.
Norwich midfielder Lee Croft then put over a teasing centre for on-loan forward Antoine Sibierski to head against the inside of the Southampton post with goalkeeper Kelvin Davis beaten.
After Southampton took the lead they almost added a second when Joseph Mills set substitute Bradley Wright-Phillips free and his 20-yard shot clipped the top of the crossbar.
After the break, McGoldrick crossed to Robertson who hit a first-time shot just over the bar.
Davis came to Southampton's rescue twice in a minute on the hour-mark with two fabulous saves.
First he denied Omar Koroma when he was clean through on goal and then brought a stunning second save from the striker after the resulting corner fell to his feet eight yards from goal.
After the sending off and penalty, Southampton looked to go on the rampage to kill the game off.
Lallana, McGoldrick and then Lallana again all went close to adding a third but it was not to be.
The win was Southampton's third of the season and they are now four matches unbeaten as they look to climb the table.