Marco Silva's side moved into eighth place, one point behind seventh-place Wolverhampton Wanderers, who host Fulham on Saturday.
Everton continued their impressive recent form, starting brightly and taking the lead after 18 minutes through Ben Mee's own goal, with the defender deflecting in Richarlison's powerful effort.
Seamus Coleman doubled Everton's lead two minutes later, heading home the loose ball after Tom Heaton could only parry Lucas Digne's drive.
Burnley striker Ashley Barnes had a goal correctly disallowed for offside as Sean Dyche's men, safe from relegation, threatened briefly.
Matej Vydra was denied a penalty when former Burnley defender Michael Keane appeared to push him over in the box and Everton held on to record their eighth clean sheet in 10 Premier League matches.
Goodison Park warmly applauded Everton's players on a lap of appreciation, reward for their impressive form in 2019.
If league leaders Manchester City win the FA Cup, the team that finishes seventh in the standings will qualify for the Europa League.
"We've ended the season well, at the mid-point of the season we were very poor," Coleman told Sky Sports.
"For periods of the season the fans were frustrated with how we were playing and that's understandable.
"We wanted to finish in seventh, maybe top six and going forward that's what we've got to aim for. This club needs a trophy and with this manager and this set of players we have to aim for it."
(Writing by Rik Sharma, editing by Pritha Sarkar)
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