The Australian quartet of Glenn O'Shea, Alexander Edmondson, Michael Hepburn and Alexander Morgan completed the 4,000 metres at the wooden velodrome in the Belarus capital in three minutes 56.751 seconds to clinch the gold medal.
However, the result was well short of the world record of 3:51.659, set by the British at the London Games last August.
Britain only had two riders, Steven Burke and Ed Clancy, returning from London, with newcomers Samuel Harrison and Andrew Tennant replacing Geraint Thomas and Peter Kennaugh. The new quartet could only manage the time of 4:00.967 in the final.
Denmark beat Spain in the consolation final to win the bronze in 3:59.821.
Earlier, American Sarah Hammer won her fifth world title in the women's individual pursuit over 3,000 metres while France's Francois Pervis clinched his first in another non-Olympic event - the men's one kilometre time trial.
Hammer crushed Australia's Amy Cure in the final by more than eight seconds, coming home in three minutes 32.050 seconds, with another Australian Annette Edmondson taking the bronze.
Hammer, who also won in 2006, 2007, 2010 and 2011, did not compete at last year's championships in Melbourne. The 29-year-old, who lost the Olympic gold medal in the omnium in London by a single point, said she did not plan a big celebration.
"I'm just going to call my parents at home, that's about it. I'll have a day's rest tomorrow, then get ready for more races on Friday, Saturday and Sunday," she said.
Pervis won the time trial, which was dropped from the Olympic programme after the 2004 Games in Athens, in one minute 00.221 seconds, with Simon Van Velthooven of New Zealand finishing second, 0.648 seconds behind, and German Joachim Eilers taking the bronze.
British team newcomer Kian Emadi just missed out on a medal in an event that six-time Olympic champion Sir Chris Hoy once dominated; Emadi's time of 1.01.756 was just over three tenths of a second slower than Eilers.
"I was stressed out, staying at the hotel and waiting for the championships to start. But finally, I'm a world champion," said 28-year-old Pervis, who was not selected for the French Olympic team in London.
"I hope to win another medal in keirin on Friday. I was second in 2009 but I'll try to go for the gold this time."
Germany won the last gold medal on offer on the opening night of the five-day championships, taking the women's team sprint title by edging out China in the final by just three hundredths of a second. Britain finished third ahead of Australia as Becky James produced a brilliant ride as she partnered teenager Vicky Williamson to bronze.
James, a double European junior champion when she last competed in Minsk four years ago, is due a busy weekend in the absence of the injured Jess Varnish and retired Victoria Pendleton.
And the 21-year old sprinter is clearly in top form, as Germany retained the title they won in Melbourne, edging out China in a repeat of last year's Olympic final
"I'm in the best form but this medal is still so unexpected, I'm shocked really, totally gobsmacked," said James, who saw her Olympic dreams ruined by an untimely bout of appendicitis.
"We came in with no expectations because we're a new line-up but I knew my form was coming through and to get on that podium is so encouraging.
"I've got plenty of events to come and after all the things that went wrong last year, it's just great that things are going right now.
"My training has been so consistent and that's the simple reason for my form. I've not raced this well since the Commonwealth Games in Delhi. .