Romelu Lukaku scored what turned out to be the winner in the 20th minute of Saturday's match as he fired home on the rebound after his initial header from Ashley Young's cross was kept out by Fraser Forster.
Jose Mourinho's charges were made to sweat for the three points, however, with Southampton improving in the second half, in one instance having Marouane Fellaini clear an Oriol Romeu header off the line.
Jones told MUTV: "lt felt like for the majority of the second half we couldn't seem to get out [of our half].
"Then when we did get the ball we gave it away too cheaply which you know caused them to come straight back at us.
"It was just back and forth, like a yo-yo, like a basketball match for all the second half but we showed character, resilience and fight. We dug in and got the three points."
United sit second in the Premier League table on 16 points, with neighbours Manchester City ahead only on goal difference.