Austin and Wallace scored in a first half dominated by the Clarets and, even though Blackpool improved drastically after the break, Bartley's late third sent the Seasiders packing.
The first chance of the game came in the fifth minute when Alex Baptiste escaped down the right and fizzed a cross towards Jonjo Shelvey, but the on-loan Liverpool man was left kicking at fresh air after mistiming his shot.
The Clarets' first sniff of goal came after 13 minutes when Ian Evatt inexplicably tried to keep the ball in on his own byline but passed straight to Austin.
The striker fired a cross in but it was well diverted wide by Craig Cathcart to spare Evatt's blushes.
But the deadlock was broken six minutes later following a spell of Burnley pressure when Wallace worked a short corner to the recalled Junior Stanislas.
Cup-tied for the midweek Carling Cup defeat at Cardiff, the ex-West Ham midfielder curled in a beauty of a cross which was met by a diving Austin to score his ninth goal of the season with a thumping header.
The hosts started the game only three points above the relegation zone and with only one home win in the league this season.
But they doubled their lead in the 27th minute after Stephen Crainey handled on the edge of his area - the Scotsman perhaps fortunate to escape with only a booking.
From the resultant free-kick, Wallace smashed a glorious left-footed strike straight into the top right-hand corner with Matt Gilks stranded.
Shelvey was another Tangerine lucky to stay on the pitch when he went over the top of the ball in frustration on Ben Mee, but was only cautioned when a red looked more appropriate.
The Seasiders, themselves with aspirations of closing in on the play-off places, were showing more urgency after the interval and finally tested Lee Grant properly after 63 minutes when Keith Southern raced clear, but Grant did well to get down to his near post and tip the midfielder's shot wide.
The visitors' pressure was on the increase - David Edgar producing a brilliant tackle to deny substitute Callum McManaman - but the Clarets sealed the points in the 79th minute.
Austin managed to break clear and square the ball for Bartley to slot home from 10 yards.
There was still time for Austin to miss a sitter, skewing wide from five yards with an open goal at his mercy, before Shelvey scored a consolation in injury time with a neat finish from the edge of the area.