Killie opened the scoring just before the half-hour mark, when Callum Booth felled Greg Kiltie in the penalty area. Jamie Hamill’s spot-kick was well saved by Scott Fox but the midfielder followed up to tuck home the rebound.
Odabeyi doubled the lead before the half-time interval with a six-yard header from an accurate Alexei Eremenko cross – the visitors capitalising on Partick’s inability to clear a corner properly.
The midfielder repeated the trick early in the second period with his ninth of the season but Partick finally gave their fans something to shout about when Kallum Higginbotham swung the ball in for Conrad Balatoni to reduce the deficit.
It proved to be a false dawn for the hosts and they conceded again in the 71st minute when Josh Magennis, who earlier had a heated head-to-head disagreement with Hamill, located the corner with a fierce strike from just inside the penalty area.
Motherwell’s defeat to St Mirren sent Kilmarnock six points clear of Ian Baraclough’s side – meaning they are safe from the possibility of having to play-off for their Scottish Premiership status.