Chelsea's win at Southampton earlier in the day had seen them overtake the pair and take top spot, but a win for either side would have seen them become the outright leaders -- and earn a significant psychological blow.
Despite earning a Premier League record 100 points last season, City lost three times against Liverpool last season in all competitions, including their first defeat of the season in the league.
But they were determined to end their dismal recent record at Anfield, where they have now failed to register a victory for 15 years -- and were presented with a glorious opportunity five minutes from time when Virgil van Dijk brought down Leroy Sane in the box. But Mahrez, who had fought of Gabriel Jesus to take the spot-kick, ballooned his effort over the bar.
Buoyed by a typically vibrant home crowd, Liverpool started on the front foot and City were struggling to match the hosts' intensity.
However, City soon began to take control of proceedings and had a strong shout for a penalty, when Dejan Lovren and Sergio Aguero tangled in the penalty area. Though neither side were able to create a clear opening.
The second half failed to liven up with both sides cancelling each other out. City left-back Benjamin Mendy was yellow carded for pulling back Mohamed Salah, when the Liverpool forward threaten to go past him, but the Egypt international was unable to exert his influence on this match.
Jesus, who replaced Aguero in the second half, worked himself an opening but was tackled by Lovren before he could execute a cutback.
The result means City go top, ahead of both Chelsea and Liverpool on goal difference, with all three teams tied on 20 points.