The British team, who collected seven of the 10 golds on offer at last year's London Olympics, made a slow start before bouncing back on the third day of the five-day event in the Belarus capital.
Yates, 20, won by a single point from Spaniard Eloy Teruel with a last-gasp spurt after trailing his rival for much of the 40-km race over 160 laps around the wooden oval of the newly-built velodrome.
"I just tried to save some energy for the end and with 10 laps to go I realised I needed just one point," he told the BBC.
"When you've got a world title on the line you get that energy from somewhere."
Olympic sprint champion Kenny then shook off the disappointment of a sixth-place finish in the team sprint on Thursday to power past his rivals on the line of the eight-lap race.
It gave Britain their second consecutive keirin title following Chris Hoy's success last year in Melbourne.
Hoy is not competing at these championships while he still contemplates his future in the sport after winning his sixth Olympic title in London.
"It was a really tough day yesterday and today as well. I had some good fortune in the semi-finals when (Francois Pervis) got relegated," said Kenny referring to the Frenchman who was disqualified for blocking the British rider.
"But I kept working hard...it's unbelievable."
German Maximilian Levy had to settle for the silver for the second year in a row, with Dutchman Matthijs Buchli taking bronze.
Earlier, Poland's Katarzyna Pawlowska retained her title in the 10-km scratch race consisting of 40 laps around the track.
"My main task was to keep everyone close enough and not let anyone break away and lap the field," said the 23-year-old who won her maiden world crown in Melbourne.
"I still felt strong coming to the final lap and made sure I had a good lead before the finish."
Mexico's Sofia Navarro came second just ahead of Yevgeniya Romanyuta who gave Russia their second bronze of the night after Kirill Sveshnikov had finished third in the points race.
American Sarah Hammer, bidding for another medal after winning her fifth world pursuit title on Wednesday, started strongly but faded midway through the race to finish second last of the 18 riders.
- Sports & Recreation
- Motor Racing