Alex Ferguson's side have now won 11 of their 12 home matches in the league this season.
But for the 12th time in the current league campaign they fell behind when Rodriguez made the most of a David de Gea error to open the scoring.
Of those 12 occasions United have only failed to win three times and Rooney summed up that character with two first-half goals that made it nine in nine matches for the England striker.
But United had to endure a strong second-half performance from the visitors and some jittery moments in the closing stages to secure their victory.
After emerging with credit from a goalless draw from his first game in charge against Everton, this was another performance full of positives for new Southampton manager Mauricio Pochettino.
United boss Ferguson could hardly have selected a more attack-minded line-up but once again his defence let him down as the league leaders fell behind after two minutes.
Morgan Schneiderlin's ball was cut out by Michael Carrick who lobbed a back-pass towards de Gea.
But the Spanish goalkeeper was weak in his challenge of Rodriguez and the striker rolled the ball into the net for an easy finish.
The lead was to last just five minutes, with Shinji Kagawa clipping through an incisive pass for Rooney to get in behind and guide a shot past Artur Boruc to pull the hosts level.
They should have moved ahead just over a minute later when a Phil Jones cross was allowed all the way across the box and Kagawa was only denied by the post.
Southampton refused to sit back and a Rickie Lambert free-kick glanced off the wall and wide before Gaston Ramirez also thumped a free-kick well off-target.
But the hosts were soon threatening again and after Robin van Persie cut in from the right his shot was deflected just over the bar by Maya Yoshida.
Just before the half-hour, United did get their second goal when van Persie floated a free-kick to the far post, Patrice Evra headed back across goal and Rooney touched in from less than a yard out.
Lambert headed over from a Schneiderlin corner at the other end before Danny Welbeck almost ended his goal drought when he calmly collected a van Persie pass only to be denied by Boruc.
Pochettino made a double change at the interval as he looked to spark his team and Lambert forced a decent save from de Gea.
Lambert also thumped wide from just outside the area as Southampton continued to grow in confidence after the break.
Steve Davis also drove well wide as the Saints pressed and de Gea had another anxious moment when he fumbled a Lambert free-kick only to grab the ball at the second attempt.
Ferguson threw on defenders Rio Ferdinand and Rafael as he looked for some stability and Rooney finally showed some attacking impetus as he had a low drive held by Boruc.
After the Poland keeper pulled off a fine save from van Persie, the Dutchman had a header ruled out for offside.
De Gea then went some way to redeeming himself for his early error with a fine save from Lambert's curling free-kick.
Boruc was also out quickly to prevent Rooney from scoring a third but Nemanja Vidic pulled off a fine block from Lambert before Schneiderlin drove well over as Southampton refused to yield.