Stand-in captain Jordan Henderson gave Liverpool the lead in the first half with his second goal in as many games, driving home a superb strike from the edge of the area, and then turned provider for Daniel Sturridge in the second half.
The win – which extends an unbeaten run in the Premier League to 12 games – keeps the Reds two points off the Champions League spots after fourth-placed Manchester United left it late to beat Newcastle.
Sturridge replaced Lazar Markovic for just his second league start since returning from injury, and it was the 25-year-old who had the game’s first chance.
After playing a nice one-two with both Adam Lallana and Raheem Sterling on the right, the England international’s left-footed shot was superbly tipped wide by Tom Heaton.
It was then Henderson’s turn to force Heaton into a save, seeing his 20-yard drive kept out well by the Burnley stopper low to his left.
But Henderson – wearing the armband with Steven Gerrard recovering from injury – got his goal on 29 minutes, firing home an unstoppable 25-yard shot into the corner after Philippe Coutinho’s effort from just inside the area had been blocked by Jason Shackell.
Coutinho went close to doubling Liverpool’s lead moments later, but his curling effort from 25 yards was just a yard wide of the goal.
Heaton made another key save just before half-time, blocking Sturridge’s low effort after he was put through on goal by Coutinho.
After the break, the ball fell to Henderson 35 yards out on the right, and the in-form Liverpool skipper found Sturridge superbly at the far post, allowing the striker to head past Heaton from close range.
Danny Ings had Burnley’s best chance of the game, doing well on the byline to dummy past Henderson, but his stretched effort was blocked.
Lallana and Alberto Moreno missed half-chances in a quieter second period, but Liverpool eased through the final periods to record their fourth straight Premier League win.
Liverpool: Mignolet (6), Can (7), Skrtel (7), Lovren (6), Lallana (6), Henderson (8), Allen (6), Moreno (6), Sterling (6), Sturridge (7), Coutinho (7).
Subs: Johnson (5), Toure (5), Lambert (5).
Burnley: Heaton (7), Trippier (6), Keane (6), Shackell (5), Mee (6), Boyd (5), Arfield (5), Jones (5), Kightly (6), Barnes (6), Ings (6).
Subs: Wallace (5), Vokes (5), Jutkiewicz (5).
Man of the match: Jordan Henderson
Watch Wednesday’s goals on Sky Sports News HQ from midnight.