Owen's poorly- struck penalty was saved by Antti Niemi, and a string of other missed chances meant Liverpool failed to capitalise on defeats for Birmingham and Newcastle, two of their rivals for fourth place.
This was the perfect start for Southampton's new manager Paul Sturrock, who admits he lives every kick of a football match. In his first game as a Premiership manager, he paced his dug-out like an expectant father.
The men who delivered twin goals to delight him were James Beattie and Kevin Phillips, both looking a great deal more threatening than the subdued and apparently disinterested Owen.
Houllier said: ' I'm not worried about Michael. He's going through a difficult period, but that is when he needs his manager to be there for him and I can tell you that he will be the first name on my team sheet for the next game.'
It has been a bad week for Houllier. Whatever support he enjoys from within the Anfield boardroom, ultimately it is the fans who will bring the pressure to bear and those who had swapped the Mersey for the Solent yesterday were hurrying for the exits when Phillips scored the second goal five minutes from time.
Yet in the first half, Liverpool were really the only side in the game, denied a comfortable half-time advantage by Niemi, who has arguably been as consistent as any keeper in the Premiership.
His early stop from Harry Kewell was outstanding. Steven Gerrard, who must despair his driving industry is producing so little reward, should then have scored from Kewell's defencesplitting pass but curled his shot beyond the far post.
And Owen was denied just before half-time by Niemi when he was put through superbly by Kewell, driving the ball against the keeper's legs where he would have dinked it over them if his confidence were higher.
Sturrock admitted: ' We didn't really deserve to win but there will be plenty of times I sit here and complain because it was us who deserved to win when we haven't.'
His joy was uncontained when Beattie scored the first after exchanging passes with Phillips, though he did look a good two or three yards offside, a belief that fuelled the touchline fury of Houllier's assistant Phil Thompson.
Even then Liverpool should have equalised, Gerrard again surging into the penalty area to be denied by the remarkable Niemi, and when Owen lunged at the loose ball he saw it come back off the post.
Niemi then denied Kewell again as Liverpool wondered what they had to do to beat him.
There were no such problems for Phillips at the other end as he collected the ball from Anders Svensson, though his shot looped off the boot of John Arne Riise to dip over the head of Jerzy Dudek and make sure of a famous victory.
