The Saints made it nine wins out of 10 by claiming a 3-2 victory against Stoke City, which has seen them book a tie against Sheffield United in the next round.
"We seem to be picking up away draws but I am delighted to get through tonight," Davis told BBC Sport. "They made it difficult for us and we are delighted to be through to the next round.
"Every game we play we want to win. We have good competition throughout the team and it is a good place to be at this point of time. We now look forward to Sheffield United."
Southampton have never won the League Cup in their history, but were losing finalists against Nottingham Forest in 1979.