It was a good start from the Blues with a chance for Eden Hazard before Pedro opened the scoring in the seventh minute. That was canceled out by Brady's magnificent effort, before a good chance for Matthew Lowton looked to make two before half-time.
The Clarets started the second half brightest with Andre Gray. The half declined from there, with half-chances from Diego Costa,Cesar Azpilicueta, Cesc Fabregas and Pedro but couldn't break through Sean Dyche's side.
Antonio Conte's side looked to increase their gap at the top of the Premier League table, after closest rivals Tottenham losing 2-0 to Liverpool the previous day and the Blues started bright at Turf Moor.
Their first knock at the door came in the sixth minute, when Costa came short to collect the ball before playing it through to Hazard. The ball caught defender Michael Keane upfield as the Belgian broke into the area, but it was a weak effort as it nestled into the hands of Tom Heaton.
They didn't have long to wait until they found the lead the minute later, and it was a great team effort from the visiting side. Hazard did well to spread the ball to Victor Moses, he did well to skip past the overeager challenge from debutant Brady, playing the ball to Pedro on the edge of the area. The Spaniard had an excellent first touch to get into the area, before he swept it past Heaton and into the bottom corner.
Making yourself known
There was some pressure on Brady after securing his club record move from Norwich City, up until the 24th minute he had a poor debut but managed to turn it around as he equalised in spectacular style.
He stood over a free-kick 25 yards out, and he did magnificently to curl it over the wall and into the top left-hand corner, leaving Thibaut Courtois absolutely no chance.
Dyche's side began to turn the screw from that point, and they came close to grabbing another as the clocked began to tick down towards the break.
The recently returned Joey Barton played an excellent ball to the full-back Lowton, who did well to wriggle his way past the mini man-mountain N'Golo Kante. He only had Courtois to beat and Gray waiting in the wings for the seemingly easy tap-in, but he decided to go alone and his effort was saved low by the Belgian keeper.
Game fizzling out
Many wouldn't have expected the scoreline as the sides came out for the second period, but it was the Clarets that started brightly in the second period.
Their opportunity came two minutes after the restart, as Ashley Barnes played a ball into the area looking for Gray but it looked to be covered by David Luiz. However the Brazilian allowed the ball through his legs, Gray picked it up and looked to beat Courtois from 18 yards but the keeper was down low to save.
After that opportunity however the game fizzled out, with only half chances arising as the sides made feeble attempts to try and claim the three points.
Ben Mee nearly gifted one to the opponents in thr 53rd minute, he allows pressure on him from Costa and gave it away but managed to recover and block the chance.
Azpilicueta then went for broke just after the hour mark, as the Spaniard tried one from 25 yards but it was well wide of the mark.
Fabregas and Pedro then had half chances towards the end, but Courtois almost had a clanger in extra-time.
The ball comes to the Belgian but he makes a hash of the clearance, it comes up in the air but luckily for him it rolls off the roof of the net.
Burnley's fantastic home record remains, while Chelsea's lead at the top remains at 10 points.