Jelavić began his career with Croatian side Hajduk Split of the Prva HNL in 2002, before moving to Belgian Pro League club Zulte Waregem in 2007.

Jelavić was born in Čapljina, SR Bosnia and Herzegovina, SFR Yugoslavia and started his youth career at his local side GOŠK Gabela, which plays in the Bosnian league system.

His career took a more serious turn when he moved to the nearby cross-border club in Croatia, NK Neretva from the town of Metković, managed by Branko Gutić.

At the encouragement of Ante Prco, Ivan Gudelj, Dražen Mužinić and Vilson Džoni, Jelavić was brought to Hajduk Split's junior side at the age of 15.

In the 2008–09 Austrian Football Bundesliga season, he played 24 games (as first squad in nine and as substitute in 15) and scored seven goals without any yellow or red cards.

[11] On 11 September, Jelavić scored his first Rangers goal against Hamilton Academical at New Douglas Park after he headed in a Kyle Lafferty cross.

[12] On 21 September, Jelavić played against Dunfermline Athletic in the League Cup, scoring two goals in the game including an overhead kick in a 7–2 win for Rangers.

After making his comeback off of the bench, Jelavić started his first match since his injury on 26 January 2011, scoring away from home against Hibernian in the first half.

[20] Jelavić scored his first European goal for Rangers on 3 August 2011 in a 1–1 draw against Malmö FF in a UEFA Champions League qualifying game.

[24][25] He made his debut a few days later, as a second-half substitute for Denis Stracqualursi, in a 1–1 draw against Wigan Athletic at the DW Stadium.

[35] In 2013–14 the loan signing Romelu Lukaku was preferred to Jelavić, who did not register a goal until January when he scored twice against Queens Park Rangers in the FA Cup third round.

[36] Though Everton manager Roberto Martínez was reluctant to sell the Croatian, Jelavić sought a move to Hull City in order to try and play more first team football ahead of the World Cup.

He finished his first season at Hull with four goals from 16 Premier League appearances, but both he and strike partner Shane Long were cup-tied for the team's run to the 2014 FA Cup Final.

In Hull's maiden European season Jelavić returned from Croatia's 2014 World Cup campaign early due to the sale of his strike partner Shane Long.

[41][42] He made his West Ham debut on 19 September, coming on as a 69th-minute substitute for Manuel Lanzini in a 2–1 away win against Manchester City.

[48] On 1 March 2021, Lokomotiva coach Jerko Leko revealed that Jelavić and teammate Sammir decided to end their professional careers, and that the club respected their decisions.

[51] On 14 October 2009, he made his first competitive international appearance for Croatia, in the team's final 2010 FIFA World Cup qualifier against Kazakhstan in Astana, again coming on as a substitute for Klasnić, in the 77th minute.

[57] In the opening match of the 2014 World Cup, on 12 June against the hosts Brazil in São Paulo, he mishit a cross from Ivica Olić which was deflected by opponent Marcelo for the first goal of the tournament.