The Reds were made to work in the first half despite having most of the ball, with both sides having their fair share of chances to go ahead.
Liverpool appeared to lack their shooting boots as chances went begging for all of the front three, as well as Thiago and Nathaniel Phillips from close range.
But the Reds finally made their mark just before the break as Roberto Firmino got on the end of Andy Robertson's ball into the box and found the back of the net to make it 1-0 at half-time.
Liverpool got into their groove early in the second half and it took them just over five minutes to double their advantage via an unlikely source.
Sadio Mane's lofted ball into the box following a Reds corner was met by the head of Nathaniel Phillips, who expertly turned the ball home to make it 2-0 to the Reds.
Burnley, spurred on by their supporters, continued to look threatening as the half wore on. But the combination of Phillips and Rhys Williams did well to resist their advances.
James Milner and Alex Oxlade-Chamberlain came on in the final minutes and it was the latter who finally put the game to bed with a well-taken finish after initially doing well inside the box and creating space to get the shot off.
The result means Liverpool are now back in the top four on goal difference - for the first time in three months - and should they win against Crystal Palace, they should have more than enough to secure Champions League qualification ahead of Leicester.