Christian Benteke's majestic 77th-minute header looked like giving Liverpool the win, but Sadio Mane equalised for the visitors to frustrate the Anfield crowd.
Klopp's arrival, following the sacking of Brendan Rodgers, was greeted with great expectation but three consecutive draws have served as a reality check.
"There was still 10 minutes to go when Southampton equalised," Klopp told reporters. "We didn't give up physically, it's not that we stopped our game physically, but we didn't believe any more that we could turn the game around.
"That's a problem, that's why we're not calm enough in the moments when we have the chances, the last pass and so on.
"I don't understand this pressure at the moment, but the guys feel it. You can see that. They work so hard, they are full of concentration, full of readiness, full of passion, everything is there.
"We conceded one goal and it felt like the end of the world -- it's not the end of the world. It's only a goal, you can always come back and that's what we have to understand."
Missing injured striker Daniel Sturridge, Klopp turned to expensive signing Benteke at halftime, and the Belgium international responded to head in James Milner's cross.
But with the home crowd preparing to celebrate, Mane bundled in an equaliser at the far post.
The Senegal international marred his day when he was sent off for two yellow cards in quick succession late on but his team held on to stay one place ahead of Liverpool in eighth position.
"We can work with this result," Klopp said. "We can work with this performance of the team. If I sit here and have to say 'these guys are not ready for fighting', then it would be bad news. But everybody saw how ready they are."
Southampton manager Ronald Koeman said his side had deserved their point.
"It was not our best performance but we had a good reaction after going down to a great goal from Christian Benteke," he said. "Our defensive organisation is good.
"Teams like Liverpool, if they have a change in management, there is always a change in expectation. I know that, I have played for big teams and I know the expectation on the manager of Liverpool is always bigger than that of Southampton."
(Additional reporting by Martyn Herman; Reporting by Steve Tongue; editing by Clare Lovell)
