Tottenham Hotspur, beaten by United on penalties in last season's final and winners in 2008, were eyeing a hat-trick of finals after they beat Everton 2-0.
Blackburn Rovers beat Championship strugglers Peterborough 5-2, Portsmouth enjoyed a rare win with a 4-0 home success over Stoke City and Aston Villa beat Sunderland 3-1 on penalties after a 0-0 draw.
United, whose first team were beaten 2-0 by Liverpool in the Premier League on Sunday, settled early nerves by taking a sixth-minute lead when Danny Welbeck, who scored the winner against Wolverhampton Wanderers in the last round, headed in from a corner.
Two seasons ago Barnsley beat Liverpool and Chelsea en route to the FA Cup semifinals and the Yorkshire side, struggling at the wrong end of the second division table, went close to an equaliser when Daniel Bogdanovic headed against a post midway through the first half. But Owen, enjoying a rare start for United, showed his predator's instinct to settle it after an hour with a sharp turn on the edge of the box to wrong-foot two defenders followed by a precise finish into the corner.