But they left it very late to earn a replay, with Martin Paterson grabbing the vital equaliser three minutes from time as the former Scunthorpe striker took his tally for the season to 16 goals.
Premier League bottom club Albion had no complaints as they always found the Clarets difficult opposition to contain, particularly in the last 25 minutes.
It had, however, appeared that the sharp shooting K-men, Robert Koren and Do-heon Kim, had knocked Burnley out of their second cup competition within the space of four days.
Albion's two foreign imports are not known for their goalscoring exploits, but on this occasion they hit the high notes to give the Midlands club hopes of a win.
But Burnley, who have already beaten three top-flight teams this season, displayed skill, commitment and passion and once again gave themselves a chance of fresh cup glory.
They might well have won as they had several good goalscoring opportunities in the second half.
Robbie Blake, Joey Gudjonsson, Paterson and Wade Elliott will all have nightmares wondering how they missed gilt-edged chances in a sparkling second half.
But in 87th minute Patterson atoned when he glided in a close-range shot to earn a replay.
Surprisingly, the tempo for a cup tie was half-paced in the early stages. Albion's emphasis on playing good possession football lacked a directness to pose any problems for the Championship side.
Roman Bednar was the only Albion player to get a shot in on target in the opening 20 minutes, but the angle was too narrow and he fired straight at the former Albion keeper Brian Jensen.
The frustrations of their midweek defeat against Tottenham, when they came within three minutes of a Wembley final, were not evident in Burnley's early play and they created a big surprise when they went ahead in the 24th minute.
Referee Mike Dean, who has a record of nine red cards and 83 bookings this season, amazed most people on the ground when he awarded Burnley a penalty for a foul by Albion skipper, Jonathan Greening, on Blake.
Veteran defender Graham Alexander, who has over 100 goals to his credit on his travels around the divisions, confidently converted the spot-kick for his sixth goal of the campaign.
This unexpected setback certainly livened up a lacklustre Albion. Their tempo quickened and they were back on level terms seven minutes after Alexander's strike.
Greening's corner was only half cleared by Clarke Carlisle to the unmarked Koren to hammer a 25-yard shot through a packed defence into the roof of the net for only his second goal of the season.
But Albion looked far from secure at the back as was highlighted in the second half. Goalkeeper Scott Carson failed to hold a low shot and only the woodwork prevented Blake from regaining the lead for Burnley ten minutes before the break.
Albion, however, scored their crucial second goal just before half-time when Kim saw his lofted free-kick deflected into the net by the unfortunate Christian Kalvenes to give the Korean his first goal of the season.
With the Clarets chasing the equaliser they left themselves a little exposed at the back as highlighted when Bednar cut across Jensen only to see his shot on the turn cannon off the post.
They went close earning a replay occurred when Blake's weak header from Alexander was easily saved by the ideally positioned Carson.
The England goalkeeper then displayed his international skill when he majestically turned away a Gudjonsson header from Blake's cross.
With Albion on the ropes Paterson wasted a glorious opportunity with a very weak shot on the turn.
Elliott then found himself with only Carson to beat, but somehow screwed his shot wide in the 78th minute, but he sent Burnley's fans home happy when he created the cross from which Paterson notched the equaliser.