Steve Cotterill's side dominated from the outset at Ashton Gate and it did not take long for Aaron Wilbraham put them ahead with his fifth goal of the season.
Right-back Mark Little made it 2-0 from close range and Rovers had Harry Forrester sent off before Kieran Agard opened his account for City with a late piledriver.
The result lifted Cotterill’s side one point above previous leaders Peterborough United, who were held to a goalless draw by Notts County at London Road.
The visitors were indebted to Roy Carroll, the 36-year-old former Manchester United goalkeeper, who made a string of fine saves; one in particular to deny Kyle Vassell.
Barnsley's five-match unbeaten run was brought to a dramatic end as they were consigned to a 5-3 hammering by MK Dons at Oakwell.
Samir Carruthers scored one goal and had a hand in two others as the Dons raced into a 5-0 lead at Oakwell and although Barnsley rallied with late goals from Conor Hourihane [two] and Devante Cole, the damage had already been done.
Eoin Doyle scored a hat-trick to take his goal tally to the season to seven as newly-promoted Chesterfield beat Scunthorpe United 4-1 to stay level on points with the Dons - who occupy third spot.
Crewe Alexandra got their first points off the season with a 2-1 derby win over Port Vale, while Colchester United and Walsall secured their first league victories of the campaign by beating Leyton Orient 2-0 and Preston North End 3-1 respectively.
Coventry City beat Yeovil Town 2-1 to make it two wins out of two since returning to the Ricoh Arena, while Swindon Town won by the same scoreline at Bradford City.
Fleetwood Town fell to seventh following a 1-0 loss to mid-table Crawley Town and Jamal Campbell-Ryce scored the only goal late on as Sheffield United edged out Rochdale.
Oldham Athletic and Gillingham played out a goalless draw in the other game to be played on Saturday.
