Jamaica international Austin latched on to El-Hadji Diouf's excellent through-ball just after the hour-mark and swept the ball home first time.
A second straight win lifts Leeds further away from relegation trouble and marks an impressive start for former Reading boss McDermott, who replaced Neil Warnock on Friday in time to take charge of Saturday's derby win over Sheffield Wednesday.
Some of Leeds' football, particularly in the second half when they showed more of a cutting edge, belied the gloom that had enveloped Elland Road when Warnock departed following four straight defeats 16 days ago.
Burnley were by no means over-run, but in the absence of leading goalscorer Charlie Austin created little in front of goal and are now looking nervously over their shoulders after slipping to within three points of the relegation zone.
Leeds twice went close to taking a deserved lead when defender Jason Pearce's header from Stephen Warnock's cross was blocked on the line by Burnley skipper Jason Shackell and Luke Varney headed the rebound on to the crossbar.
Burnley served warning of their capabilities with a strong finish to the first half, first Ross Wallace shaving the foot of the post with a drilled effort from outside the penalty area and Martin Paterson lashing an angled drive wide.
Varney, Leeds' two-goal matchwinner on Saturday against Wednesday, spurned a golden chance within seconds of the restart as he steered Diouf's right-wing cross off target.
Varney was then denied by Burnley goalkeeper Lee Grant, who tipped his far-post header from Ross McCormack's free-kick over the crossbar.
Leeds fans were audibly appreciative of the fare on show as their side continued to pass the ball with patience and purpose and they were duly rewarded in the 62nd minute. Diouf showed fine technique to control McCormack's cross-field pass and clipped a perfectly-weighted ball over the top for Austin to run on to and convert with a first-time side-footed finish.