The Clarets defended well though and almost grabbed a late winner when Ben Mee had a header cleared off the line.
Liverpool, with Philippe Coutinho starting his first Premier League match of the season, created the first opening after only three minutes when Daniel Sturridge’s 15-yard effort was easily gathered by Nick Pope, who was replacing the injured Tom Heaton in the Clarets goal.
The Reds had penalty appeals waved away by referee Roger East in the 20th minute when Mohamed Salah went down under James Tarkowski’s challenge.
Coutinho’s 20-yard shot was then deflected over the bar by Mee and Salah was unable to direct Sturridge’s cross on target as Jurgen Klopp’s side continued to dominate.
However, Sean Dyche’s side took the lead with their first shot on goal in the 27th minute when Scott Arfield drilled a low shot past Simon Mignolet.
Liverpool responded well though and grabbed an immediate equaliser when Salah controlled the ball superbly before firing a low-left footed drive past Pope - the Egyptian’s fifth goal in eight appearances for the Reds this season.
The home team went close to scoring again with Sturridge only just failing to get a touch to James Milner's cross before Salah almost set up Coutinho.
However, Liverpool continued to look fragile defensively and Mee was only just off target with a ten-yard header in first-half stoppage time.
The Reds continued to press in the second period and Sturridge and Coutinho went close to making it 2-1 as Burnley defended doggedly.
Milner’s volley was then blocked by Tarkowski as the wave of Liverpool attacks continued and Emre Can headed a presentable chance over Pope’s goal.
However, the visitors almost scored when Mee’s header looped over Mignolet only for Joel Matip to guide the ball over the crossbar.
Pope then made a terrific stop to keep out Trent Alexander-Arnold's late effort and Dominic Solanke smashed a volley on to the top of the crossbar but Liverpool were unable to grab a last-gasp winner.