Despite having little to play for other than pride, Burnley, who are now in 17th place, responded to the return of fans at Turf Moor with an aggressive start to the game.
Chris Wood struck the side netting from a promising position and the Clarets were first to the ball across the field in the opening exchanges.
But Liverpool wasted two good opportunities with Thiago Alcantara and Mohamed Salah both off target after finding space in the box.
Burnley winger Dwight McNeil brought a good save out of Liverpool keeper Alisson Becker with a 25-metre drive but it was the visitors who made the breakthrough.
Sadio Mane fed Andy Robertson on the left and his low cross was expertly slotted home by Firmino, the Brazilian looking confident again after his two goals at Manchester United last week.
Phillips eased any lingering Liverpool nerves with a firm header at the back post from a Mane cross from the left but he had to be alert at the other end to clear a Ben Mee header off the line as Burnley looked for a way back into the game.
A fine individual effort from substitute Oxlade-Chamberlain wrapped up the win in the 88th minute, the former Arsenal midfielder twisting to find space for a fierce low drive past Will Norris.
Juergen Klopp's side will face Crystal Palace at Anfield on Sunday while Chelsea are at Aston Villa and Leicester are at home to Tottenham Hotspur.
"We feed off pressure. It's a massive club and fans demand results," said Robertson.
"We have hit form at the right time. Sunday is a final effectively for us, Leicester and Chelsea. We have given ourselves a chance when a lot of people had written us off. But we need to make sure our performance is of the highest level (against Palace)."
Champions Manchester City and second-placed Manchester United have already secured Champions League qualification.