Steve Cotterill's side took advantage of Swindon Town not playing due to the international fixtures to enhance their title credentials at Ashton Gate, with Burns securing the win in the third minute of stoppage time.
Chesterfield had twice pegged their hosts back through Eoin Doyle, the striker taking his tally to 14 goals in 11 league games with a double to cancel out Ian Evatt's own goal and Derrick Williams' strike, but they could not hold on as City recorded an eighth win of the league campaign.
Peterborough United climbed into second place after coming from behind to beat Crawley Town 4-1.
After Izale McLeod had given the hosts the lead early on, Peterborough found their feet and they were level after 11 minutes when Michael Smith fired home from the centre of the penalty area.
Christian Burgess put Darren Ferguson's men ahead for the first time just before the break with Marcus Maddison and Conor Washington completing a comfortable victory.
At the bottom, there were morale-boosting victories for Crewe Alexandra and Scunthorpe United.
Febian Brandy struck his first goal for Crewe on 26 minutes to give the strugglers an early lead against Coventry City, with George Cooper doubling their advantage six minutes later.
The two-goal cushion at the break gave Steve Davis' men confidence and although Anthony Grant put through his own net three minutes after the break, Crewe held on to record only their second win of the season.
Coventry's challenge was effectively ended when Jack Finch was sent off for a second bookable offence late on.
Scunthorpe, who sacked manager Russ Wilcox on Wednesday, also enjoyed victory on Saturday as caretaker manager Andy Dawson guided them to a 3-0 win at Gillingham.
With new manager Mark Robins not set to be officially appointed until Monday, Scunthorpe were given an edge when Doug Loft was sent off after 34 minutes for Gillingham and the visitors took the lead four minutes before half-time through Paddy Madden.
They ultimately ended a run of four straight defeats in all competitions thanks to goals from Neal Bishop and Gary McSheffrey in the second half.
As the bottom two were collecting three points, the sides immediately above them were also recording wins to move out of the bottom four.
Port Vale's 4-1 win over Yeovil Town, that included a Tom Pope double, saw them climb into 19th place, while Colchester United jumped five places to 16th after coming from behind to beat Fleetwood Town 2-1.
Oldham Athletic also had to put the disappointment of an early goal behind them to beat Walsall 2-1.
Meanwhile, at Bramall Lane, Sheffield United thought they had secured a dramatic win over Leyton Orient with two stoppage time-goals but Romain Vincelot's 99th minute strike saw the match finish 2-2.
