Jaidon Anthony gave Burnley a 37th-minute lead at Hillsborough and Josh Brownhill doubled the advantage shortly after half-time.
Victory extended Scott Parker's team's unbeaten run to seven matches and moved them two points above Sunderland, who play Hull on Sunday.
Blackburn maintained their 100 percent start to the season in the league at Ewood Park with a 1-0 win over Swansea thanks to Tyrhys Dolan's 13th-minute strike.
Rooney, serving a one-match touchline ban, watched from the stands as his side were hammered in Cardiff.
Ibrahim Cissoko received a straight red card late in the first half to compound the former England captain's misery- his side are yet to win away this season.
"We got really sloppy, we gave the ball away a lot, we didn't win tackles or do the basics right," said the Manchester United great.
"In this league, if you don't do that right, you have difficult days like today.
"It's difficult watching it from the stands, you want to be there at the side of the pitch trying to help the team."
Freddie Potts and Callum Lang struck to give Portsmouth their first win of the campaign as they came from behind to beat QPR 2-1.
Emil Riis scored in the 72nd minute as Preston beat Coventry 1-0 while substitute Dane Scarlett headed a dramatic injury-time equaliser to earn Oxford United a 1-1 draw against West Brom.
There were also wins for Luton and Bristol City while Millwall drew 1-1 against Derby and Stoke played out a 1-1 draw against Norwich.
