Ten Hag named the same starting line-up that defeated Liverpool 2-1 on Monday, meaning unsettled star Cristiano Ronaldo and club captain Harry Maguire were both on the bench.
Ronaldo, keen to leave Old Trafford before the transfer window closes on 1 September, was largely anonymous when he came on for the last 20 minutes.
Casemiro helped United close out the win as the Brazil midfielder made his debut in a 10-minute substitute appearance following his move from Real Madrid.
United were some way short of the intensity and quality that inspired their win against Liverpool.
But they showed the grit so badly lacking in the first two games as they kept Southampton at bay after Fernandes' goal.
Ten Hag will be pleased that Raphael Varane and Lisandro Martinez stood firm at centre-back, rewarding his decision to keep Maguire on the sidelines.
It took an incredible combination of superb goalkeeping and last-ditch defending to stop United taking the lead midway through the first-half.
When Southampton failed to clear a cross, Gavin Bazunu had to scramble to block Anthony Elanga's shot at the far post.
In the ensuing melee, Christian Eriksen looked certain to score but Kyle Walker-Peters blocked his shot on the line before Fernandes saw his goalbound strike repelled by Armel Bella-Kotchap.
Bella-Kotchap should have made United pay for those misses when a corner fell to the Saints defender, but he somehow contrived to shoot high into the stands from close-range.
Che Adams couldn't generate enough power to beat David De Gea after making a perfectly-timed run behind the flat-footed United defence.
Scott McTominay's drive tested Bazunu early in the second half, while Bella-Kotchap made a superb tackle to stop Elanga as he prepared to shoot after bursting through on goal.
Although United were never dominant, they conjured an impressive move to break the deadlock in the 55th minute.
A flurry of passes scythed through the Southampton midfield and Diogo Dalot curled a cross towards Fernandes, who guided a fine volley past Bazunu from just inside the area.
Southampton pushed for an equaliser but they were frustrated when a strong penalty appeal was rejected by VAR after McTominay handled as he challenged Adams for a high ball.
De Gea preserved United's slender lead with a reflex save from Joe Aribo's close-range header, with Walker-Peters nodding the rebound over.
Bella-Kotchap blazed over and Sekou Mara shot wide as Southampton wasted their chance to deny United victory.