But Saints' enterprise was rewarded in a penalty-box scramble, when Lovren got his foot to the ball and prodded it goalwards, although skipper Adam Lallana might also have got the last touch from a position right in front of David de Gea.
It means United have won only three times from their eight league games so far and the gap to pace-setters Arsenal has now widened to an imposing eight points.
And it had all looked so good too, with Adnan Januzaj celebrating his new five-year contract with a hugely impressive display.
It was Januzaj who claimed the assist for Van Persie's goal after providing a sensational 50-yard pass to send Wayne Rooney racing through.
Artur Boruc was out quickly to deny the England man, only for Van Persie to seize on the rebound and net his seventh goal of the season.
Januzaj also brought a fine save out of Boruc with a thunderous second-half effort, then released Danny Welbeck to create what would have been a second United goal had Morgan Schneiderlin not made a magnificent covering tackle.
The value of that became evident at the end as, with Sir Alex Ferguson watching on front the stands, Moyes' search for the winning formula continued.
An upbeat mood prior to kick-off was partly due to that precious win at Sunderland a fortnight ago, but also news the youngster whose goals secured victory had committed his long-term future to the club.
Yet it obscured the sobering reality of United starting the game in ninth spot.
That Southampton dominated the opening 20 minutes did not bode well.
Mauricio Pochettino had taken the bold step of leaving Rickie Lambert out of his starting line-up in favour of Jay Rodriguez and was rewarded with an enterprising start in which the former Burnley man twice went close with headers.
Victor Wanyama also tried his luck before the visitors' best opportunity arrived when Rooney coughed up possession to Lallana close to his own box.
The Saints skipper quickly squared to Dani Osvaldo but his effort lacked conviction and De Gea, maintaining an ever-present record, made a comfortable save.
In response, Van Persie had gone close with an audacious first-time effort from Jonny Evans' lofted pass and also had a shot saved by Boruc after Nani had provided the assist.
United were in search of inspiration. Januzaj provided it.
Moyes knows the tradition of young players being given a chance at United. It seems he has got one from the very top drawer. Certainly United were desperate to tie the 18-year-old down to a long-term deal.
And in the brilliant pass that released Rooney, Januzaj showed why.
Boruc was out quickly to make a brave block. But Van Persie reacted equally swiftly.
He nipped around the prone goalkeeper before drilling home his first league goal from open play since the win at Swansea two months ago.
United nearly doubled their advantage soon afterwards when Nani created space for himself inside the area, then rolled a pass into Rooney's path which the in-form striker crashed against the crossbar.
Southampton were not retreating into their shells though.
De Gea needed to watch Lovren's 30-yard shot closely before he gathered and Evans needed to be alert to prevent Lallana sliding in to finish off a flowing move by touching home Steven Davis' low cross.
The introduction of Lambert should have given the visitors even more impetus, yet it was United who responded better.
In a frenzy of penalty-box action, Van Persie guided a header against the crossbar, then, after Evans had returned it to the six-yard box, Nani controlled and smashed the ball goalwards, only for Boruc to make a splendid save.
A minute later, Januzaj, who had both been booked and been the victim of a tackle which caused Wanyama to be cautioned, let fly from 25 yards.
His vicious shot dipped and swerved, but Boruc still managed to get a hand on it and Marouane Fellaini skied the rebound over.
Januzaj's influence remained strong. It was his touchline pass that released substitute Danny Welbeck, only for Schneiderlin to make a magnificent challenge before the United man could set Rooney up with a tap-in.
It was Southampton who had the last word though, with De Gea saving from Nathaniel Clyne before that mad, scrambled leveller.