The hosts went closest to snatching a sixth consecutive home league win, with Matthew Lowton and Andre Gray both wasting presentable opportunities.
The result means Chelsea, who face Wolves in the FA Cup fifth round next Saturday, sit 10 points clear at the top of the table, although Manchester City can cut that to eight if they beat Bournemouth on Monday Night Football.
Meanwhile, 12th-place Burnley move onto 30 points, having enjoyed another creditable home performance against one of the Premier League's top teams. They have now gone six top-flight games unbeaten in front of their own supporters for the first time since September 1975.
Sean Dyche made two changes from their defeat at Watford, handing full debuts to Ashley Westwood and Brady.
Antonio Conte kept faith with the team that dismantled Arsenal at Stamford Bridge and the Blues made a flying start once again.
Victor Moses burst down the right and teed up Pedro, who kept his composure to finish a brilliant Chelsea counter-attack.
It was just the start the visitors wanted in front of a boisterous Turf Moor crowd and they looked set to push on before Brady struck against the run of play.
The Ireland international, who was caught out of position for Chelsea's opener, made amends with his equaliser, giving Thibaut Courtois no chance with a 25-yard free-kick into the top corner.
It sparked the hosts into life and with heavy sleet coming down, they squandered two fantastic chances to take the lead either side of half-time.
Their first opening was created by the impressive Joey Barton, whose fine pass picked out the advancing Lowton only for the full-back to fire straight at Courtois.
And two minutes after the break, Gray did likewise when mistakes by Gary Cahill and David Luiz put the Burnley striker one-on-one with the Chelsea goalkeeper.
Luiz, who wore heavy strapping around his right knee dating back to his clash with Sergio Aguero at the Etihad, was tested time and again by Gray, whose tireless running kept Chelsea's defence on their toes.
With Burnley on the front foot and their front two causing problems, Conte ditched his wing-back system in the second half and went to a back four.
He introduced Willian and Cesc Fabregas in an effort to re-establish control of midfield and it largely worked.
The Clarets sat back in the closing stages and looked to hit their opponents on the counter, but ultimately both sides seemed happy to settle for a point.
In fact, Pedro's seventh-minute goal turned out to be Chelsea's final shot on target of the match; a testament to Burnley's rugged defensive display.
Burnley: Heaton (7), Lowton (7), Keane (8), Mee (8), Ward (7), Boyd (7), Barton (8), Westwood (7), Brady (8), Gray (7), Barnes (7)
Subs used: Vokes (6), Arfield (6)
Chelsea: Courtois (7), Azpilicueta (7), Luiz (6), Cahill (6), Moses (7), Kante (7), Matic (6), Alonso (7), Pedro (7), Hazard (7), Costa (7)
Subs used: Fabregas (6), Willian (6), Batshuayi (6)
Man of the match: Michael Keane