Naby Keita bagged Liverpool's fastest ever Premier League goal before Sadio Mane and Mohamed Salah also got on the scoresheet.
Liverpool now hold a two-point lead over City, but the defending champions have a game in hand.
Liverpool and Bayern Munich played out a cagey 0-0 draw in the first leg of their Champions League last-16 tie at Anfield on Tuesday.
The Senegal international scored 10 goals as Liverpool reached last season's Champions League final.
Mohamed Salah has scored 50 goals for Liverpool quicker than any other player in the club's history after just 65 appearances for the Reds.
Manchester City remain top of the table on goal difference, but both teams are now locked with Chelsea.
Jurgen Klopp put Firmino on the bench but brought him on with about 20 minutes to go.
Mohamed Salah was accused of diving by Crystal Palace when he tumbled in the penalty area late in the first half at Selhurst Park.
Mohamed Salah got Liverpool up and running as they comfortably dispatched West Ham in their first Premier League fixture of the season.
World Cup 2018, Senegal vs Colombia Preview: Sadio Mane Key To Senegal's Hopes As They Meet Colombia
Senegal and Colombia would be hoping for a Round of 16 berth from Group H.
Both Senegal and Colombia are strong contenders for a Round of 16 slot from Group H.
Senegal coach Cisse hopes that Mane can be the difference, on what promises to be a nerve-wracking final round of matches on Thursday.
FIFA World Cup 2018, Highlights, Poland vs Senegal: Senegal beat Poland 2-1 to register Africa's first victory of the tournament
Poland and Senegal will begin their campaigns in World Cup Group H.
Liverpool's Sadio Mane hopes to return to Bambali, where his family still lives, with a winners medal.
Klopp's men started their Champions League campaign in a playoff against Hoffenheim just to make the group stages back in August.
Liverpool, inspired by two-goal Egyptian Mohamed Salah, took a big step towards the Champions League final with a 5-2 win over AS Roma in their semi-final, first leg at Anfield on Tuesday.
Mohamed Salah and Sadio Mane were on the scoresheet as Liverpool ran out comfortable winners against Newcastle United.
