Kieran Trippier opened the scoring for the visitors before David Norris levelled matters and Portsmouth remained on top until the interval, giving little indication of what was to follow.
Danny Ings netted his first goal for the Clarets early in the second half before substitute Austin claimed a quickfire treble - his final two goals coming in stoppage time.
Pompey went into the match buoyed by two consecutive home wins but are now five points from safety with six games remaining.
Their magnificent second-half showing gave Burnley a first win in eight games and Eddie Howe's side remain 16th in the table.
Portsmouth boss Michael Appleton named the same side that beat Hull 2-0 on Tuesday night.
Teenage midfielder Cameron Howieson was named on the bench for Burnley, who were also unchanged.
Following a quiet opening, the game sprung into life when the visitors went in front with 16 minutes gone.
Dean Marney's low corner from the left flank was dummied to Trippier and the full-back sent a sweetly-struck drive into the bottom corner from 25 yards.
Portsmouth's response was swift and clinical. Luke Varney scampered onto George Thorn's searching pass down the right wing and cut the ball across goal for Norris to thump into the roof of the net.
Appleton's men were now on the front foot and Scott Allan fired narrowly over the angle of post and bar after Burnley defender David Edgar carelessly coughed-up possession.
But the Clarets almost took the lead on the stroke of half-time when Ben Mee's strike took a deflection off Tal Ben Haim and forced Jamie Ashdown to make a superb save.
They were back in front two minute after the restart courtesy of Ings' measured glancing header from Edgar's right-wing cross after Portsmouth only half cleared Trippier's corner.
Austin was introduced in the 53rd minute when fellow forward Martin Paterson appeared to suffer a hamstring strain - his latest blow in an injury-hit season.
Varney headed wide from Kelvin Etuhu's cross as Pompey searched for an equaliser, but a staggering collapse followed.
Ross Wallace's 74th minute free-kick caused confusion inside the hosts' penalty area and Austin was on hand with a predatory finish.
Josh McQuoid fired over after a driving run from Ings and Ashdown was forced to palm a goalbound Chris McCann cross to safety before further gloss was added to the scoreline in stoppage time.
An under-hit backpass from Portsmouth centre-back Jason Pearce gifted former Swindon man Austin his 50th career goal and Junior Stanislas and McQuoid engineered a sweeping move for him to complete his hat-trick from close range.