Oliva, part of the British trio that finished third in the team sprint 24 hours earlier, progressed to the final the hard way, coming through the repechages after he was penalised for entering the sprinters' lane in the first round.
However, the 21-year old was on the right end of track officials decisions in the final, taking third after Canada’s Hugo Barrette was relegated and Malaysia's Azizulhasn Awang and Germany's Joachim Eilers were disqualified.
Matthijs Buchli, of the Netherlands, and Colombia's Santiago Ramirez won gold and silver respectively.
In the women's omnium, London 2012 team pursuit gold medallist Dani King lies fourth after the three events, 18 points behind Australia's Isabella King in bronze medal position.
And British team-mate Jon Dibben, 19, is also fourth in the men's event, which concludes with the individual pursuit, scratch race and kilo on Sunday.