Michael Appleton watched his team dominate the first period but four second-half goals sunk Pompey who are now five points away from npower Championship safety with six games to go.
However, Appleton is convinced survival is attainable and predicts it may well go to the last day of the season.
Appleton said: "It is going to be tough but I still believe that we can do it. We all do. We have to.
"I do not mind if it it goes down to the last day as long as we stay up. I am still confident but it was a big blow to lose today. I felt we dominated the first period but they came back into it and were a lot better than us in the second half. I was disappointed with that."
Portsmouth were looking for their third home win in a row and started brightly with Tal Ben Haim's misguided cross forcing Lee Grant into a good save at his near post. It was Burnley who opened the scoring though after 16 minutes when Dean Marney fired a low corner to the edge of the area to Kieran Trippier who blasted a shot in from 20 yards out. Pompey equalised just three minutes later when Luke Varney cut a cross back to David Norris who slid home from close range.
The home side continued to push forward with Chris Maguire coming close with a angled drive from eight yards after being put clear by Ben Haim and Scott Alan shooting just over from 18-yards out. Eddie Howe's side almost snatched the lead again moments before the break but Jamie Ashdown did well to tip over Ben Mee's deflected looping shot.
Two minutes into the second half Burnley retook the lead when David Edgar crossed for Danny Ings who flicked a header into the bottom left-hand corner for his first goal for the club.
Relegation-threatened Pompey pushed forward for another leveller and came close through Varney who met substitute Kelvin Etuhu's cross but his header went inches wide.
After that Charlie Austin stole the show with a sublime hat-trick first slotting home on 73 minutes to make the game safe. A quick-fire double in the 90th minute from former Swindon striker Austin made it a rout and helped his side push for a late play-off place. Burnley manager Howe believes he has the best strikers in the Championship after five stunning goals against Portsmouth.
Howe praised: "I think we have the best strikers in the Championship. We have Charlie, Jay, Danny who are all top quality forwards. They are all very clinical and they showed that. Portsmouth dominated the first half but I thought we were excellent in the second period."