The Reds appeared to be heading for a welcome win - and Klopp's first since taking over at the start of the month - when Christian Benteke headed home with 13 minutes remaining.
However, Sadio Mane equalised nine minutes later to earn Saints a point and afterwards Klopp admitted his side needs to react better to setbacks.
"It's only a goal, only a game of football. It's like it's the last thing in your life and we have to calm this down," he said.
"I don't understand this pressure, but the guys feel it. I hope I'm not the only person in the stadium who thought 'this is not the end of the world'. We can work on this."
The German, who will hope to secure his first win in charge when Bournemouth visit Anfield in the Capital One Cup on Wednesday, says the side is progressing the way he wants even if results are eluding them.
"I saw development, I saw the next step, but at the end we are disappointed because of the result," added the former Borussia Dortmund coach.
"The problem is the other teams don't wait for us - they want to win, they want to beat us."
"Football is not a fairytale. Sometimes we write stories like this but today it didn't happen. We can change a lot of things if it's necessary, but not all things will work from the start. If you work on something you are not 100% better the next day."