Goals from David Norris and Connor Wickham gave Town the points in an entertaining game at Turf Moor.
Norris struck just past the quarter hour from close range for his 10th goal of the season and then played a big part in the second, his cross headed in by Connor Wickham before the break.
Jay Rodriguez pulled a goal back for the home side with a low drive from outside the box but Town held on to make it a third game on the road without defeat.
Town boss Paul Jewell made one change from the team that beat Scunthorpe before the international break, the suspended Lee Martin replaced by Jason Scotland. Kieron Dyer, back after a thigh injury, started on the bench.
Ross Wallace had the first chance of the game on 10 minutes, the Scotland international firing wide after Dean Marney had set him up.
Arran Lee-Barrett was quickly off his line to clear his lines from outside the box, his opposite number Grant doing the same at the other but in more spectacular style, heading the ball into touch.
Damien Delaney was in the right place to intercept Rodriguez when the Burnley frontman looked set to go clear on goal.
Scotland had a shot blocked on 17 minutes for a corner but from the flag kick, Blues went ahead. Gareth McAuley headed the ball back across goal and Norris made no mistake with a left foot finish from 10 yards.
Scotland could have added a second soon after, Jimmy Bullard's free-kick was met by the Trinidad international at the nearpost but he volleyed over.
Scotland was blocked out again on the edge of the box before Town extended the lead on 25 minutes, Norris dummied the ball and then spun off for the lay off from Scotland, the skipper's cross headed in by Wickham at the far post - the ball bouncing before evading Grant and going into the roof of the net.
Chris Eagles threatened to reduce the arrears but was pulled up for handball inside the Town box, Wallace then booked for a foul as Burnley's frustration grew.
The former Sunderland midfielder made way for Chris Iwelumo on 38 minutes in what looked a tactical change by Clarets' boss Eddie Howe.
The final five minutes of the half brought a flurry of yellow cards, with Wickham, Delaney and Norris going into referee Trevor Kettle's book, Andre Bikey also booked for a foul on Norris on the edge of the box, Bullard sending the free-kick straight at Grant.
Lee-Barrett held firm from Marney's strike from 18 yards after Iwelumo's head down as Burnley looked for an early reply in the second-half, Iwelumo going down under Mark Kennedy's challenge and looking for a penalty, Nathan Delfouneso then having a shot blocked as the Clarets started brightly.
Lee-Barrett got his angles right to deny Delfouneso, the 'keeper then joining the growing list of bookings for Blues for timewasting.
Wickham just failed to get on the end of a terrific Carlos Edwards' cross after good link up play down the right, Bullard then sending a left foot effort high and wide.
Wickham had two efforts blocked as Town threatened to seal the points on the hour mark, Scotland crowded out at the far post as he looked to convert from Bullard's corner.
Burnley made a second change with Wade Elliott on for Eagles for the last 23 minutes, and a minute later the home side were back in the game, Rodriguez finding the bottom corner with a superb drive from 20 yards.
Town could have extended the lead inside a minute though, Norris denied by Grant's fingertips after a flowing move involving Edwards and Scotland.
The entertainment continued with Rodriguez stabbing the ball just wide from a free-kick, Town then making a first change with Andy Drury on for Scotland for the final quarter hour. Burnley also making a swap with Graham Alexander on for Marvin Bartley.
Delaney bravely cleared from Elliott inside the box, Delfouneso then booked for a foul on Lee-Barrett.
Dyer replaced Josh Carson for the closing stages, Drury then lashing the ball over the bar from 25 yards but Blues held for the points to take the tally to 52 and dent Burnley's Play-Off hopes.
Burnley: Grant, Mears, Duffy, Bikey, Easton, Eagles (Elliott 67), Bartley (Alexander 78), Marney, Wallace (Iwelumo 38), Rodriguez, Delfouneso. Subs not used: Jensen, Duff, Carlisle, Edgar. Booked: Wallace, Bikey, Delfouneso.
Town: Lee-Barrett, Edwards, McAuley, Delaney, Kennedy, Norris, Leadbitter, Bullard, Wickham, Carson (Dyer 80), Scotland (Drury 75). Subs not used: Fulop, Peters, Healy, Civelli, O'Dea. Booked: Wickham, Delaney, Norris, Lee-Barrett.