Despite conceding a huge amount of possession to their English rivals the Hoops, with five serious matches behind them thanks to Champions League qualifiers, succeeded where it mattered most - in front of goal
Amido Balde scored with their first chance of the game and while they spent a long time defending they did it effectively even though Liverpool created enough chances to have won the game themselves.
Brendan Rodgers' side had gone into the game on the back of six friendly victories, conceding just once against Valerenga on Wednesday, but they lacked a cutting edge.
The return of Daniel Sturridge, out with an injured ankle since May, for 45 minutes in the second half, could not have come soon enough with the Premier League season only a week away.
With last season's leading scorer Luis Suarez suspended for six matches even if he remains at the club - Arsenal are still keen despite owner John Henry adamant they will not sell to a rival - and summer signing Iago Aspas still looking lightweight up front there is a real need for someone of Sturridge's ability to provide the spearhead.
Stewart Downing, whose future is again in doubt after interest from West Ham, provided some positive moments but it may have come too late to save his Anfield future.
Philippe Coutinho produced some flashes of brilliance but also failed to threaten goalkeeper Lukasz Zaluska, who was kept busy in the first half against a tide of attacking red shirts.
But the most significant impact was made by Balde, who gave young centre-back Andre Wisdom, deputising for vice-captain Daniel Agger who was left on Merseyside, a tough time for over an hour before he was substituted having done the hard work.
In the 13th minute the Portuguese striker easily turned Wisdom after being put through by Anthony Stokes and drilled in a low shot which caught the underside of Simon Mignolet's body but bounced up and over the line.
Downing almost equalised immediately with a shot which was blocked while Coutinho latched onto Glen Johnson's through-ball only for Zaluska to save.
The goalkeeper was also required to palm Downing's shot around a post after the winger had cut in from the right and he also denied Johnson.
With Liverpool enjoying a vast amount of possession Celtic defended resolutely and sought to hit their opponents on the counter-attack but their only other chance came right at the end of the half, with Stokes' header from Emilio Izaguirre's cross failing to test Mignolet.
Sturridge was sent on for the second half in place of Aspas and within three minutes he did what the Spaniard had failed to do in 45 by producing a threat with an early shot.
Coutinho somehow managed to completely miss Downing's cross and shoot wide having created an opening for himself either side of Sturridge forcing a near-post save from Zaluska.
Late on, Sturridge's lob from another inch-perfect Coutinho pass drifted wide of the far post and a fierce strike from the edge of the area rippled the side-netting.
The last thing Liverpool wanted with Stoke to face in next weekend's curtain-raiser was for doubts to creep in about their capabilities up front in the absence of Suarez, but Sturridge may yet provide the antidote if he can persuade Rodgers he is fit enough.