It's now four games unbeaten ahead of crucial Premier League clashes against Brighton and West Brom in the next seven days.
“We are really pleased with many aspects," Saints boss Mauricio Pellegrino said.
“One of them is that we had the opportunity to progress to the next round, and we did it.
“On the other hand, we are also growing as a team and we are improving – this is the feeling of the group.
“In the next week we have some important games, and I think it is the best moment for us to approach those games. For those reasons, we are happy."