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.
The visitors' 28-goal leading scorer Charlie Austin had emergency surgery to remove his appendix on Tuesday morning after falling ill on Monday and will miss the rest of the season.
Burnley midfielder Chris McCann's curling effort and Leeds striker Ross McCormack's volley were all either side had to show for an enterprising first 25 minutes.
But Leeds twice went close to taking a deserved lead in the 30th minute 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 Matt Paterson lashing an angled drive wide.
Varney, Leeds' two-goal match-winner 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 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.
Burnley responded through McCann's low long-range effort and substitute Danny Ings volleyed wide soon after.
McCormack almost made it 2-0 with a blistering free-kick that whistled inches wide and although Leeds looked anxious at times in the closing stages, Burnley could not capitalise.