The Saints took the lead in the third minute when Carrick’s poor back pass allowed Rodriguez to intercept and finish past the stranded David De Gea.
Wayne Rooney’s first-half brace put the Red Devils back in control at Old Trafford, ensuring the 19-time English champions moved seven points clear of rivals Manchester City.
“Michael’s pass-back was a bit soft and Michael held his hand up – its great to see that, honest players like Michael saying ‘Sorry boss, sorry lads’. It’s great,” said Ferguson.
United were far from convincing in the second half as Rickie Lambert and Jay Rodriguez both tested De Gea with free-kick efforts.
And Mauricio Pochettino has sympathy for United’s Spanish goalkeeper after the 22-year-old’s difficult start to his Premier League career.
“It is clear to me that any free-kick near to the box, or a corner, presents a goalscoring opportunity,” he said.
“That doesn’t happen as much in the Spanish league.
“Also, the goalkeeper in Spain is a bit more protected than here. That makes their job here a lot harder.”
Meanwhile, Ferguson believes United’s victory could prove vital in the Premier League title race after a dogged Red Devils display saw off plucky Southampton.
“In the second half Southampton produced the best performance anyone has done here this season,” he added.
“They pushed right on top of us and didn’t give us time to settle.
“But my experience of this club, when you are going for championships, is that there are always games where you can say we were a bit lucky. This was one of those nights.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news