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 on penalties after a 0-0 draw.
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.
"It was a Cup tie, Barnsley have put a lot of effort in and had one or two chances when we went down to 10 men but I think we deserved to get through," Ferguson told Sky Sports.
Picking out Owen's contribution he said: "It was a fantastic goal, good feet, quick and a marvellous finish, I'd say it was a great goal."
Spurs went ahead after 31 minutes with a superb counter- attack ending with Tom Huddlestone's powerful strike.
Stand-in goalkeeper Brad Guzan was Villa's hero at the Stadium of Light as he saved a penalty from Kenwyne Jones to force extra time before keeping out three Sunderland efforts in the shootout to help his side to a 3-1 penalties success.
