Nunez's precision finish after six minutes gave Liverpool a flying start at Turf Moor but Juergen Klopp's side were unable to kill off the hosts as the second goal would not come.
Mohamed Salah was denied by the crossbar and both Cody Gakpo and Harvey Elliott had goals disallowed as Burnley hung on but Diogo Jota's 90th-minute effort sealed the points.
Nineteenth-placed Burnley had chances of their own in the second half and almost snatched an equaliser when Jacob Bruun Larsen's shot went agonisingly wide of the post.
Liverpool's first win in three league games moved them above Arsenal as they reached the halfway point of the season on 42 points. Arsenal, who have 40, host West Ham United on Thursday.
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