The result has seen the Toffees move above Brentford into 15th spot in the Premier League table, four points ahead of 18th-placed Luton Town, while Burnley stay 19th, six points from safety.
The victory is Everton's first in England's top flight since the middle of December, when they recorded a 2-0 success over Burnley at Turf Moor.
Everton came close to an opener in the early exchanges, with Dwight McNeil's wind-assisted corner causing Burnley a problem, but the visitors managed to survive the uncomfortable moment.
Smart goalkeeping from Arijanet Muric prevented Calvert-Lewin from potentially having a tap-in from an Abdoulaye Doucoure cross in the 16th minute, with the hosts on the front foot.
There was frustration inside the stadium as the match ticked towards its 30-minute mark, with Burnley looking dangerous on the transition, and smart defending from James Tarkowski was required to clear a cross from Lorenz Assignon, who was back in the side following a suspension.
Jacob Bruun Larsen had the chance to test Everton goalkeeper Jordan Pickford from a free kick in a promising area in the 34th minute, but his effort was just too high.
An Andre Gomes free kick late in the half caused some panic inside the Burnley box, but the danger was ultimately cleared, adding to the frustration inside the stadium.
Everton managed to make the breakthrough in the closing moments of the opening half, though, with Calvert-Lewin scoring following a horror moment from the Burnley goalkeeper.
Muric advanced with the ball before attempting to deliver down the field, but instead he smashed the ball straight into Calvert-Lewin, and it bounced into the back of the net to hand Everton the lead.
Everton defender Jarrad Branthwaite made a brilliant block to deny Sander Berge early in the second period, with the Clarets still looking dangerous in the final third of the field, but they were shaky at the back.
Calvert-Lewin had another opportunity after poor defending from Vincent Kompany's side, but Muric was on hand to make a smart save to prevent it from becoming 2-0.
Burnley were reduced to 10 men in the 64th minute of the match, with Dara O'Shea given his marching orders for a foul on McNeil, who was advancing through on the away side's goal.
The Clarets came close to a leveller shortly after, with substitute Josh Brownhill turning inside the Everton box before looking to pick out the top corner, but his effort was just wide.
Branthwaite again made an excellent challenge to prevent Josh Cullen going through on goal, with Kompany's team threatening to level despite being down to 10 men.
Beto had a chance to score a second for Everton late on, but his effort was blocked, and one goal ultimately proved to be enough for Sean Dyche's side to pick up all three points.