However, whilst Wayne Rooney's hot streak continued with a first-half brace that took him to nine goals in nine games, of far more concern to Sir Alex Ferguson was his side's shocking display after the interval, and David de Gea's goalkeeping.
The visitors spent almost the entire second period camped in the United half and were only denied a deserved point by some desperate defending as United failed completely to stem the tide of attacks.
Despite their lofty status, certain negative topics have gone round on a depressing loop for the Red Devils this season. Bad defending, weak goalkeeping, conceding the first goal.
So, after just two minutes came the perfect storm. An underhit and misdirected back pass from Michael Carrick, De Gea failing completely to block, and Rodriguez stroking into an empty net.
As the goal came not that long after news emerged of a Stoke bid for England keeper Jack Butland, stories linking Asmir Begovic with an Old Trafford switch were instantly gaining strength.
On this occasion, De Gea should not shoulder all the blame.
Indeed, if he had 'cleaned out' Rodriguez, as some in the stands suggested, he could quite easily have conceded a penalty and been sent off, leaving his team-mates with a much stiffer task than the one they ended up facing.
Equally, it cannot be denied De Gea's attempt to intervene was soft in the extreme and offered more ammunition for those who say the Spain youngster, as agile as he is, simply cannot cope with the demands made of him in the Premier League.
Southampton's problem was that despite their delirium, they still had another 88 minutes to get through against a side so used to coming from behind it is almost routine.
They never looked likely to manage it.
One excellent through-ball from Shinji Kagawa was all it took to slice through the visitors' defence, allowing Rooney to charge forward and score.
Kagawa hit a post on United's next attack and it took some excellent goalkeeping from Artur Boruc to keep the hosts at bay as returning Robin van Persie and Danny Welbeck both threatened danger.
Rooney's first goal had been his 100th at Old Trafford.
It also meant Manchester United have scored in a record 61 consecutive Premier League home matches.
In neither instance was it likely to be the last.
And when Patrice Evra climbed highest at the far post to meet Van Persie's free-kick, the France defender left Rooney with the easiest of tap-ins from barely a yard.
Not that anything can be taken for granted with Sir Alex Ferguson's side, as the United boss himself outlined in his programme notes.
And with the hosts struggling to find their rhythm at the start of the second period, Mauricio Pochettino's team pushed forward in search of a leveller, with substitute Lallana having a significant influence as Rickie Lambert twice went close.
Amid the kamikaze defending that eventually heralded the introduction of Rio Ferdinand, perhaps the most telling moment came when De Gea spilled Lambert's long-range free-kick.
A gasp went round Old Trafford before De Gea pounced to deny Lallana a tap-in, sarcastic cheers - admittedly from a minority - when the keeper next caught the ball indicating the former Atletico Madrid man may be reaching the point of no return with the Red Devils.
Briefly, United relieved the pressure, with Van Persie's goalbound header brilliantly saved by Boruc before the Dutchman had an effort ruled out for a debatable offside.
But Southampton were soon back on the offensive, which meant, finally, United fans were able to cheer something positive from De Gea, an excellent flying save to turn away Lambert's free-kick.
Nemanja Vidic flung himself in the way of another Lambert piledriver before Rooney, who finished playing on the left as defensive cover for an overrun Evra, ensured the Saints' final assault got nowhere.