Born in Mostar in Bosnia and Herzegovina, Puljić spent most of his career in Croatia, first passing through the ranks of Hajduk Split's academy, and them joining second-level side Zadar on a free transfer in the summer of 2006. After establishing himself as a first team regular and helping the club win promotion to top level at the end of the 2006–07 season, Puljić went on to spend the following five seasons at the club making 106 first league appearances for Zadar.[1]
In July 2011 his contract at Zadar ended and he signed for Dinamo Zagreb's farm team NK Lokomotiva and after only one appearance for Lokomotiva in the 2010–11 Prva HNL season, he was transferred on a year-long loan to Dinamo in August, but was returned to Lokomotiva in January 2012.[1] On 23 December he signed a contract binding him to KAA Gent till 2017, starting 3 January 2014.[2]