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.
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 semi-finals 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.
United then had experienced full-back Gary Neville sent off for a high tackle on Adam Hammill and they had to work hard in the latter stages as Barnsley made, and wasted, several decent chances.
"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."
Tottenham and Everton fielded near-full strength teams and produced a lively game and a bizarre goal at White Hart Lane.
Spurs went ahead after 31 minutes with a superb counter- attack ending with Tom Huddlestone's powerful strike.
Robbie Keane doubled the lead after 57 minutes in a mighty scramble featuring no less than four rebound attempts after his penalty had been saved.
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.
The fourth round concludes on Wednesday when Arsenal face Liverpool, Chelsea play BoltonWanderers and Manchester City take on second division Scunthorpe, the last remaining lower league club.
