The visitors took the lead in comical circumstances with only three minutes on the clock thanks to Rodriguez.
However, United were level minutes later when Rooney placed the ball into the far corner following Shinji Kagawa's clever pass.
Kagawa then saw his effort on the angle strike the post as the hosts pushed hard for a second goal.
They were rewarded for their persistence on 30 minutes thanks to Rooney's simple finish after Patrice Evra had headed Robin van Persie's free kick back across the box.
Lambert had United worried on the hour as De Gea failed to hold his free kick. However, there were no Southampton players to latch onto the rebound.
The hosts were well below their best in the second half, but they defended their lead to see them move further ahead of rivals City.