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 a Delfouneso 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 saved Ross Wallace's swirling effort.