A day after winning the Olympic team pursuit with his British team mates in world-record time, Clancy rode his lap in 12.556 seconds, more than half a second faster than second-placed Shane Archbold of New Zealand.
Australian Glenn O'Shea was third.
The omnium is a test of all-round skills over two days in six disciplines: a flying lap, 30-km points race (20 km for women), elimination race, 4-km individual pursuit (3km), 15-km scratch race (10km) and one-kilometre (500m) time trial.
The Olympic men's individual sprint event turned into a farce when Briton Jason Kenny was left with no opponent for his first round final match.
Kenny clocked the fastest time in the qualifying 200-metre flying lap used to seed the riders but, with 17 competitors at the start, the 2011 world champion had no opponent.
Second in the qualifying flying lap was Gregory Bauge, who did not even get on the bike after his Greek opponent failed to show up.
Earlier, Kenny had laid down a marker, averaging more than 74 kph to set an Olympic record of 9.713 seconds in the flying lap.
Triple world champion Bauge was second fastest, ahead of Australian Shane Perkins and German Robert Foerstemann, third and fourth respectively.
Spain's Hodei Mazquiaran Uria's saddle disintegrated in his qualifying lap but he escaped unscathed and was granted a re-run.
Seventeen riders take part in the individual sprint competition after a new International Cycling Union (UCI) rule limiting each nation to one rider per event.
It meant that on Saturday, only five of the top 10 riders from the world championships started the individual sprint event and that Beijing champion Chris Hoy of Britain cannot defend his title after being omitted in favour of Kenny