Klopp told BBC Radio 5 live: "It could and should have been 2-0, 3-0.
"The best thing for us is the result. We know that we can play better at Anfield, nothing is decided."
Nathan Redmond's first-half goal gave Southampton the win, but the former Norwich winger should have increased their lead from two good chances.
Redmond himself said after the game that he "should have scored four", after he hit the crossbar and Liverpool goalkeeper Loris Karius made several good saves.
Klopp added: "We cannot be happy with the performance, Southampton cannot be happy with the result. I can't remember a clear chance for us.
"We gave the ball away so easily and we needed Loris Karius to save our lives two or three times."
"We will strike back, we will be a different team, a different side, everything will be different at Anfield," he said.
"It's still possible to go to Wembley, and that's the target.
"We have to show a reaction - but it would be cool if we could already show a reaction at the weekend, because we probably need a better performance to get something at Old Trafford [against Manchester United on Sunday in the Premier League]."