The Clarets with this point climb to within five points of a play-off place.
Steven Thompson stepped off the substitutes' bench to head a deserved late equaliser against his former club as they trailed to Michael Chopra's opener for Cardiff in first-half stoppage time.
Boss Dave Jones said: "It was a difficult game and we probably should have seen it out, but the manner we conceded so late on was very disappointing.
"It was the only time we switched off, but they have good players and we gave the ball away far too easily during the second half.
"It was a soft goal to give away and you cannot leave the near post unattended. If it was from the centre of the goal then fair enough, but not from that position."