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The visitors made a bright start to the match with Junior Stanislas firing into the side-netting from outside the box within the first minute.
Phillips played in Kirk Broadfoot moments later, but his low shot flew just past the far post.
In the 15th minute, Stanislas ran towards the edge of the Burnley box before unleashing a shot towards goal, but a combination of Matt Gilks and the inside of the post kept the game goalless.
Two minutes later, Phillips's cross was met by Delfouneso's header in the box before dropping to Thomas Ince, but his shot was gathered by Grant at the second attempt.
After a quiet spell it was Burnley who had the next chance, but Gilks made a fantastic save from Ross Wallace's swirling effort.
The game finally witnessed an opener 13 minutes into the second half, when Sylvestre cut in from the left before curling a fabulous shot into the top corner to put Blackpool ahead.
The hosts nearly made sure of the win a few minutes from time, but Ince's effort came back off the crossbar before Sylvestre sent a free header wide.
However, Burnley failed to create any significant chances, and Blackpool held on for the win.