Adam Lallana scored one goal and made two more for Guly Do Prado and Oscar Gobern as Saints won 3-0 at St Mary`s.
"We are delighted to be in the third round and to have come through a potentially tricky tie," said Adkins. "Cheltenham are a decent side, who play good football but we did a very professional job.
"We passed the ball well, have scored three goals and kept a clean sheet so I could not ask for any more.
"It is great to be in the hat for the next round and I am looking forward to the draw. The last 14 years I have wanted to pull out Liverpool but it has never materialised, we will probably get Scunthorpe away, that would be something special."
On Lallana`s performance, Adkins added: "Adam Lallana got us off to a perfect start with the early goal and is a joy to watch. I would pay to come and watch him myself and he showed why, I thought he was excellent."