The disappointing game was petering out in a goalless draw when the Czech connected with Emile Heskey's left-wing cross.
The vital win gives Liverpool a one-point lead over Manchester United and a two-point advantage over Arsenal, who have two games in hand.
Gerard Houllier was back directing operations from the dug-out once again, but Michael Owen started on the bench following his hamstring problems.
Nicolas Anelka and Dietmar Hamann returned after missing the magnificent win over Roma as Jari Litmanen and Vladimir Smicer made way.
Chelsea, who beat Liverpool 4-0 at Stamford Bridge in December, brought in Stanic and Emmanuel Petit for Sam Dalla Bona and the suspended Graeme Le Saux.
Liverpool had won just one of the last eight league meetings between these sides and Chelsea began intent on adding to their poor recent record.
Petit fired a left foot shot wide from around 25 yards out.
Abel Xavier then fouled Eidur Gudjohnsen to concede a free-kick 30 yards out, only for Jimmy Floyd Hasselbaink to drive his shot straight into the Liverpool wall.
Phil Thompson jumped from his dugout to remonstrate with fourth official Jeff Winter after referee Mark Halsey failed to award a free-kick to Liverpool on the edge of the Chelsea area for handball by Marcel Desailly.
Not to be outshouted, Chelsea boss Claudio Ranieri then sprang to the edge of his tactical area to dispute a throw-in with Winter, whose ears must still have been ringing after Thompson's blast.
Stanic then shot a curling effort well wide of Jerzy Dudek's goal.
Liverpool had still to settle after their heroics in midweek and Chelsea were looking by far the most menacing team.
Jesper Gronkjaer escaped down the right and put in a danger ball behind the Liverpool defence, which the stretching Gudjohnsen steered just wide of Dudek's left hand post.
Chelsea were inching closer to that first goal and after Jamie Carragher had been booked for pushing over Gronkjaer on the right hand side of the box, the Blues should have scored on 26 minutes.
Petit put Gudjohnsen in behind the Liverpool defence on the left and he chipped the ball to the back post, where Stanic hit a volley which Dudek expertly parried. The loose ball fell to Hasselbaink who blasted over.
Liverpool were forced to make the first change two minutes later when Steven Gerrard came off injured to be replaced by Smicer.
Gerrard looked in some discomfort and watching England manager Sven-Goran Eriksson probably knew before being told that the midfielder will not be able to play in Wednesday's friendly against Italy at Elland Road.
Liverpool had still to threaten Chelsea's goal and Carlo Cudicini was a virtual spectator.
Emile Heskey did find strike partner Anelka in a dangerous position on the right, but the Frenchman sliced his cross behind the goal.
Celestine Babayaro became the second player to be cautioned for pulling down Heskey, while Carragher was somewhat fortunate to escape a second yellow and thus a red card for a foul on Frank Lampard.
HT Liverpool 0 Chelsea 0Liverpool needed to step up not just one, but several gears in the second half and with Arsenal's games in hand, they could not afford to let this chance slip.
Initially the second half followed the same pattern as the first with the Merseysiders unable to make many inroads on the Chelsea goal.
Anelka was having little joy up front, while on the sidelines Owen continued to warm up those trying hamstrings, knowing his introduction was looming.
Liverpool did fashion an opening when Heskey chested down John Arne Riise's throw-in, but Anelka mishit his shot well wide.
Hamann then needed treatment after he headed away Hasselbaink's driven free-kick.
Heskey went close with a glancing header from Riise's free-kick on the left before Owen duly entered the fray on 57 minutes.
The England striker came on for Xavier and went up front as Liverpool switched from 4-4-2 to the 4-3-3 which had worked so well against Roma.
It was Chelsea, though, who should have scored - but Gronkjaer shot wide from Lampard's great ball.
Chelsea continued to pepper the Kop with their shooting, and Gudjohnsen turned and fired wide.
Finally Liverpool stirred.
William Gallas halted the twisting and turning Owen in the box - and then Riise drove a left-foot shot into the arms of Cudicini, almost winding the Italian.
Hasselbaink might have snatched all three points for Chelsea three minutes from time when he shot over after playing a one-two with substitute Gianfranco Zola to unlock the Reds defence.
But then in the first minute of stoppage time, Liverpool did when Heskey crossed from the left for Smicer to volley home on the right.
FT Liverpool 1 Chelsea 0