A mistimed Virgil van Dijk challenge on substitute Leroy Sane presented the Citizens with a chance to snatch all three points in the final five minutes, only for Mahrez to blaze the ball over the crossbar.
City just about edged things over the 90 minutes in a contest that did not live up to the pre-match expectations, with the game only really coming to life in its final quarter.
A point apiece means that champions City and title challengers Liverpool join Chelsea on 20 points at the top of the table heading into the two-week international break.
Home boss Jurgen Klopp handed Dejan Lovren his first league start of the season in place of Trent Alexander-Arnold, meaning a defensive reshuffle that saw Joe Gomez shift across, while Jordan Henderson took over from Naby Keita.
City, meanwhile, showed four changes from their most recent outing, including a first start in a month for Benjamin Mendy and a surprise recall for Riyad Mahrez in place of Sane.
The Reds, boasting a formidable home record that had seen them brush aside all comers to visit here this season without conceding, started the match on top.
Mohamed Salah's dragged shot from a promising position five minutes in offered hope of an end-to-end match, but the first half did not quite pan out that way.
Instead, the two meanest defences in the division would come out on top in an opening 45 minutes that did not see a single shot on target.
Man City grew into the game and ended the first half slightly on top, but they were unable to turn possession into chances.
The Citizens' best moment came a quarter of the way into the match when Sergio Aguero was caught from behind by Lovren in the box, but referee Martin Atkinson decided against pointing to the spot on this occasion.
Liverpool, not helped by the loss of midfielder James Milner to a hamstring injury, continued to look flat well past the hour mark.
Salah, who had scored or assisted five goals in his last three outings against City, then got a weak shot on target up the other end and blasted into the stands in the space of three minutes.
Sturridge's only real opportunity to make an impact came shortly after his arrival, nodding Gomez's cross into the hands of Ederson.
City, aiming to become the first side in more than 830 minutes to breach the Reds' defence at Anfield in the Premier League, saw another half-chance come and go for Mahrez - the Algerian's shot not having enough conviction to beat Alisson down low.
There would be an even bigger opportunity for Mahrez late on after substitute Sane was felled by Van Dijk, only for his penalty to end in the stand behind Alisson's goal from the game's final - and best - chance.
MANCHESTER CITY (4-3-3): Ederson; Walker, Stones, Laporte, Mendy; Fernandinho, Silva, Bernardo; Mahrez, Aguero (Jesus 66'), Sterling (Sane 76')
LIVERPOOL (4-3-3): Alisson; Gomez, Lovren, Van Dijk, Robertson; Wijnaldum, Milner (Keita 29'), Henderson; Mane, Salah, Firmino (Sturridge 72')