The pair won the final sprint to secure silver with 31 points as Australia claimed victory with 40 points.
“It’s quite nice that we’ve got so much better that we can do and are still second,” said Barker. “I’d rather have won.”
In the men’s omnium, Britain’s Mark Stewart secured silver.
Stewart, who won the tempo race during the event’s four races, was a late call-up and finished on 109 points, with winner Benjamin Thomas, of France, accumulating 130 points.
“I got the call last night about 9pm,” said Stewart. “I was sitting watching the racing, thinking: ‘I really want to be there.'”
Britain’s Jason Kenny had to settle for fourth in the men’s keirin.
Kenny, who won his heat in qualifying for the final, made an early surge for victory but was caught after the last corner as France’s Sebastien Vigier won.
“I’m happy with my performance,” said Kenny, who was beaten in the quarter-finals of the men’s sprint on Saturday.
“Disappointed with the result a little bit but I’ve moved forward a lot and got a lot better.
“I got stuck in in the final and stuck my neck out.”