Saints found the perfect solution to their infuriating habit of letting the lead slip at home - wait until the last minute of injury time to go in front.
Trond Soltvedt showed immaculate timing to beat both the clock and the offside trap to score his first goal for Southampton with just seconds remaining.
Only nine days after the crushing despair of conceding a last-gasp equaliser against Derby, Saints revelled in the joyous freedom of knowing Liverpool had no time to hit back.
For once it was the opposition who went in front and then squandered chances to kill off the game only to lose concentration in the dying seconds.
Extra-time looked certain as James Beattie nodded down for Kevin Davies who swaggered forward with some of his old adventure visibly returning on his second Dell debut.
He chipped a glorious ball over the Liverpool defence and Soltvedt darted into the gaping hole behind a static Stephane Henchoz. For once the flag stayed down.
Brad Friedel hesitated coming off his line and the Norwegian calmly steered home left-footed to fire Southampton into the fourth round of the Worthington League Cup.
It was all the sweeter since Michael Owen of all people had squandered a golden opportunity to win the tie with six minutes left on the clock.
Jamie Redknapp slipped the ball forward for Erik Meijer to bustle past the tired challenge of Claus Lundekvam and bear down on goal from the left.
He shirked the responsibility of the shot by squaring for Owen who let it run to beat Francis Benali. At that point there was a collective groan from home fans who would have put a month's wages on the England striker scoring.
But astonishingly he clipped the base of the left-hand post and at that point Saints sensed it might just be their night.
Owen had earlier missed an equally simple chance when David Thompson put him in behind Lundekvam on 13 minutes. In trademark fashion he darted clear, slipped the ball past the advancing Paul Jones and wide of the right-hand post.
Yet he scored the most difficult of his opportunities to give the Merseysiders the lead on 53 minutes with Saints again contributing to their own downfall.
Steve Staunton's clearing header forward should have been mopped up by Dean Richards whose crossfield ball went straight to Thompson.
He found Owen and England's version of Marian Pahars skipped past Richards who stumbled as he tried to retrieve the situation. Benali got across quickly to cover and left Owen with only the tiniest gap to aim at but somehow he found the one spot which goalkeeper Jones could not reach.
Richards though redeemed his error in spectacular style to make an emphatic point to Gerard Houllier who had pulled out of a move to sign him in the summer.
The equaliser came 15 minutes later from a free-kick won by the sharp turn of Pahars past Hen-choz 35 yards out.
Matthew Le Tissier took an intelligent quick free-kick while Liverpool were still re-grouping and while the otherwise impressive Sami Hyypia was wrong side of Richards.
That forced Friedel to charge from his line but he got there too late and the big defender launched himself forward to thump home a powerful header.
It rattled the Merseysiders who until then had controlled the game in midfield where Saints missed the menacing presence of Mark Hughes. For all their industry Southampton struggled to assert themselves and were too often forced on to the back foot with Stuart Ripley and Hassan Kachloul forced deep to provide extra cover.
Pahars again chased everything in attack but got little service as Dave Jones' men found it tough going against an uncompromising Liverpool side who were clearly treating the competition seriously.
The first half was fairly even in terms of possession although the Anfield side had the better chances and almost took the lead in a frantic scramble.
Hyypia's shot was blocked by the legs of goalkeeper Jones and the half clearance by Soltvedt was looped in by the head of Owen only for the keeper to claw out from under the bar. Southampton's only real opening came five minutes before the break when Jason Dodd put Soltvedt through. He lashed wide of the near post with Le Tissier screaming for a pass but offside at the far post.
Liverpool looked in control in the third quarter of the game but then did a passable impression of Southampton by losing concentration at a set-piece and dozing off in the final minute.
That was probably just as well because some of the Saints were looking dog-tired after chasing their tails for 90 minutes. In the closing stages Dodd was becoming increasingly exposed without Ripley's protection and it was Houllier's men who looked the more likely to win it in extra-time.
But for once Saints got the breaks and though not at their best they ground out a great result only for the farcical home-away draw to instantly diminish some of the euphoria.
Converted for the new archive on 25 January 2001. Some images and formatting may have been lost in the conversion.
Comments: Our rules
We want our comments to be a lively and valuable part of our community - a place where readers can debate and engage with the most important local issues. The ability to comment on our stories is a privilege, not a right, however, and that privilege may be withdrawn if it is abused or misused.
Please report any comments that break our rules.
Read the rules hereComments are closed on this article