Gary Rowett's side were twice pegged back at Turf Moor as it finished 2-2 in the lunchtime kick-off.
Blues twice took the lead through Jon Toral and a Paul Caddis penalty but were denied all three points thanks to a late Matthew Taylor free-kick.
In the opening minutes both sides exchanged spells of possession and it was the home crowd who sprang into action after five minutes appealing for a penalty against Jonathan Spector who bundled down Lukas Jutkiewicz in the area, but referee David Coote waved away the protests.
Three minutes later and Tomasz Kuszczak was called into action, smartly saving a Michael Kightly free-kick from 25 yards out. From there, Kuszczak's long clearance downfield was nodded down by Clayton Donaldson, into the path of Jon Toral who fired his low effort past a sprawling Tom Heaton to put Blues ahead after only ten minutes.
That sparked Burnley into action, looking to peg Blues back immediatley but Kuszczak was on hand to deny George Boyd on the edge of the area and then equal to the rebound from Jutkiewicz.
Rowett's side absorbed the pressure from the Clarets in the minutes following Toral's goal and after some neat build up play forced a corner which eventually found Demarai Gray on the edge of the Burnley box, who volleyed narrowly wide after 18 minutes.
Blues began to retain possession and looked to frustrate their opponents, just after the half-way point of the opening 45 minutes and Blues showed great composure on the ball and enjoyed a silky spell of possession which nearly presented Gray with an opportunity but Michael Keane made an timely interception in his own penalty area.
Just before the half-hour mark and Blues looked to recreate the strike that gave them a tenth minute lead, long ball up to Donaldson, nodded down, but this time instead of Toral, the ball fell to David Cotterill who fired over the bar.
Moments later, the aforementioned Cotterill went even closer, when his curling left-footed strike from 20 yards came back off the upright, leaving the home side and a disgruntled Turf Moor counting their blessings that the lead hadn't been doubled.
Five minutes later and Jutkiewicz again brought Kuszczak into action, the Burnley forward reacting quickest to Kightly's chest down, but his volley was easily held by the Blues 'keeper.
Five minutes before the interval, with Burnley pushing forward, Blues nearly caught them on the counter. Goalscorer Toral leading the siege, with Donaldson and Cotterill in support he opted for Blues' number 9 who managed to force a corner, where the unmarked Cotterill seemed to have been the better option.
From the resulting corner, Burnley cleared at the near post, only as far as Stephen Gleeson, 30 yards from goal who hit a dipping effort first time which forced Heaton into a fine save, tipping over the bar.
Five minutes after the restart and Blues contunied to frustrate Burnley and created the first chance of the second half. Gray getting to the byline, his cross headed clear by Duff, only as far as Maikel Kieftenbeld whose volley fell narrowly wide.
Moments later the hosts drew level, Kightly's in-swinging free-kick was flicked on by Jelle Vossen at the front post and was finished by Michael Keane who headed home at the back post.
Just before the hour and Keane almost doubled his tally and put his side ahead. David Jones' in-swingning corner was misjudged by Kuszczak, presenting Keane with a glorious chance to turn the tie his in side's favour, but the defender headed narrowly over the crossbar.
Three minutes later and Blues were awarded a penalty. An incisive break from Blues culminated in Donaldson been caught by Tendayi Darikwa, referee, Coote pointing to the spot with no hesitation, which Paul Caddis duly fired home.
Blues continued to hold firm and looked equally as threatening on the counter attack. With 15 minutes remaining Rowett opted to sure up his midfield by introducing David Davis.
However, despite Burnley's frustration and Blues resilience, nine minutes from time the hosts pulled level for the second time. Burnley were awarded a free-kick 25 yards from goal, unfortunately for Blues, substitute Matthew Taylor stepped up and whipped his free-kick in off the under-side of the bar leaving Kuszczak rooted to the spot.
A minute after drawing level, full-back Ben Mee got to the byline and his crossed flashed across the face of goal, luckily evading the onrushing Burnley attackers.
Into the final minute and goalscorer Taylor fired wide after Marvin Sordell found himself in scoring position, the ball wouldn't sit kindly for the forward and Blues could only clear as far as Taylor, who lashed his effort wide.
The fourth official indicated five minutes of added time. With Burnley pressing hard for a winner, they were almost handed it when Kuszczak again misjudged Jones' corner, fortunately his error went unpunished.
A hard fought point for Rowett's side, who for most of the match frustrated their opponents and on another day may well have hung on for all three points.