Drawing a fourth blank against Saints this season - courtesy of Fraser Forster saving James Milner's second-half penalty - handed the initiative to Manchester United in the quest for Champions League qualification.
However, Jose Mourinho's fifth-placed side, with a match in hand, missed the chance to close the gap to two points after their 2-0 loss at the Emirates.
It means third-placed Liverpool will still qualify for the Champions League if they win their remaining two matches irrespective of what United and Arsenal, seven behind with two games in hand, manage.
"We have only one point more, we wanted to have three points more
It is not the end of the world but we are really disappointed," said Klopp.
"It is not nice to drop points at home but now we play away (at already-safe West Ham) and then we have another home game.
"It is only a mental issue if you see it as one.
"Before the game we needed three (wins), and obviously now I would say two would not be bad.
"It really makes sense that we are really concentrated on West Ham.
"They are also in a positive season-finishing mood - no pressure any more - and they will try to get results somehow.
"We are still fighting and nobody has given up.
"I spoke immediately after the game to the boys - I cannot and don't want to take away the disappointment - but what I can say is that we have one point more than before."
Milner apologised to his team-mates for his penalty miss.
"It was my fault for us not getting the three points today, definitely my fault
I take responsibility," he told Sky Sports.
"I've been around long enough not to let that worry me
I did what I wanted to do - obviously it was not good enough on this occasion.
"Fraser's a good keeper
I can't do any more than hold my hands up and apologise to the boys.
Claude Puel, who has not lost in six encounters against Liverpool as Lyon and now Southampton boss, was happy with their defensive display which saw them concede 65 per cent possession and fail to have a shot on target.
"It was a fantastic game with a strong defensive unit," he said.
"It was important to play like this against this team
It was difficult for us to make some counter-attacks but it is a fantastic work for the squad.
"We defended very well: good structure, good organisation, good discipline."
Source : PA
Source: PA