Mourinho, who was sent to the stands in the last minute of stoppage time as Southampton piled on the pressure in the closing stages, was glad to come away from St Mary's Stadium with three points after an unimpressive performance.
United held on to take its tally to 16 points from its opening six games, matching its best ever start to a league. It was something so pronounced throughout this match, especially given the slight sense that there was a result here to be taken against the pre-game league leaders.
That, however, was more down to Southampton's forward play rather than the defending of Mourinho's side.More news: Iran tests new missile despite USA warnings
It was a frustrating afternoon for Rooney, who was left with blood pouring from his face after a first-half elbow when jumping for an aerial ball with Simon Francis, but no penalty was given.
Southampton manager Mauricio Pellegrino has refused to confirm if centre-back Virgil van Dijk will start. The Red Devils are now joint top alongside rivals Manchester City with both teams tied on wins, draws, goals scored and goals conceded. But as is the case in football you will not play your best every time you walk out on the pitch. All of them can score more goals than they do.
They nearly paid the price five minutes into the second half.
Oriol Romeu and Shane Long each had an opportunity for a late equaliser, but the duo matched Southampton's woes in front of goal. United's stellar form this season is largely attributable to Mkhitaryan and Mata's attacking midfield partnership, where they dovetail and interchange on the right and down the middle of the attack.
"The quotes included "#RespectEachOther" and anti-discrimination body Kick It Out has reacted to the chants re-emerging on Saturday.
Substitute Ander Herrera missed a chance to settle United's nerves and, perhaps, the match but shot a foot high from 18 yards.