The move to Burnley was confirmed on 1 September 2014, Boyd signing for an undisclosed fee "believed to be in the region of £3million" and on a three-year contract. He made his debut in a 0–0 draw against Crystal Palace at Selhurst Park on 13 September 2014, playing the whole match. Boyd scored his first goal for Burnley in the club's 3–1 defeat to West Ham United at Turf Moor on 18 October 2014, briefly halving the deficit when he reacted the quickest to latch onto Michael Kightly's cross. He scored three goals within the space of a month in December 2014 into January 2015 when he netted twice in two draws with Newcastle United, as well as scoring the first goal in a 2–2 comeback draw away at Manchester City. Boyd proceeded to score in the corresponding home fixture against Manchester City on 14 March 2015, his fifth goal of the campaign, when he "arrowed in a low half-volley" to secure a "famous" 1–0 win. He made 37 appearances during the season, scoring five times, as his first season at the club ended in relegation back to the Championship. Boyd finished the season as having covered an average of 7.45 miles per game, the highest average distance of any player in the Premier League that season. He was voted Burnley's Player of the Year at the end of the season, which Boyd described as a "great honour".