Owen's clinical 59th-minute strike settled the issue, but the victory was marred by the dismissal of Gary Neville three minutes later for an ugly challenge on former Liverpool winger Adam Hammill.
Sir Alex Ferguson fielded a totally different eleven to the one which lost at Anfield on Sunday, with Owen, Neville, Wes Brown, Ben Foster and Anderson among the starters in a strong side which also included French debutant Gabriel Obertan.
Brothers Fabio Da Silva and Rafael Da Silva also lined up, while Barnsley boss and former United favourite Mark Robins made five changes from the weekend, with ex-United product Jon Macken leading the line.
United made a lightning start and stunned the hosts by going in front after just six minutes.
A precision near-post corner from Anderson was headed home emphatically from close range by the unmarked Danny Welbeck, who notched United's winner in the previous round against Wolves.
Soon after, Welbeck almost made it two with his rasping shot saved by ex-Old Trafford keeper Luke Steele.
Steele showed his mettle again to race out and deny Obertan with the visitors totally in the box seat.
Barnsley eventually awoke from their slumber and Daniel Bogdanovic headed over following Hammill's corner and minutes later, another flag-kick from Hammill was diverted just off target by Stephen Foster.
Macken just failed to control a throughball from Anderson De Silva with the goal beckoning, while no-one could apply the finishing touch following a dangerous cross from Bogdanovic as the hosts finished the half on top.
The momentum continued on the restart with De Silva dragging a good opportunity wide.
Sub Zoran Tosic then curled an effort just over as United sought a killer second, which duly came when Owen steered home a precision shot past Steele.
Neville then saw red and sub Iain Hume headed a chance wide following a run from the lively Hammill.
Bogdanovic, Darren Moore and Hammill all spurned opportunities to heap pressure on the visitors.
Foster then made a super save to deny sub Jacob Butterfield as the litany of missed chances started to stack up.