Cardiff were not without their chances, however, with Kenny Miller going close after 15 minutes.
Joe Mason was next to go close for Cardiff after 38 minutes, but his drilled shot was pushed away by Grant to keep the scores level at the interval.
Burnley started the second period on the attack and came close to taking the lead after 54 minutes courtesy of Junior Stanislas, but his effort was blocked by Ben Turner.
Chances continued to fall Burnley's way in the second period with Kieran Trippier coming close with an effort moments later, as the home side struggled to stem the flow of pressure.
Cardiff simply could not engage their forwards in the play in the latter stages of the match, with possession lost for the home side all too often.
Both side's pushed for the winner in the latter stages, but to no avail, as Cardiff miss the opportunity to move into the play-off places.