Stoke City edged out Tottenham Hotspur 7-6 on penalties after the game finished 0-0 at the end of extra-time and Newcastle United beat Nottingham Forest, like Spurs four-time winners of the trophy, 4-3 in a thriller after extra time.
Arsenal had to come from behind to beat League Two (fourth division) Shrewsbury Town 3-1 at The Emirates while Bolton Wanderers won 2-0 at fellow Premier League side Aston Villa.
Former England striker Owen put Manchester United ahead after 15 minutes at Championship (second division) side Leeds with a scuffed shot that trickled in off the far post.
Owen grabbed his second goal in the 32nd minute with a superb right-foot strike from just inside the area.
Ryan Giggs made it 3-0 just before the break when he cut in from the left and struck a delightful shot with the outside of his left foot.
Manchester United manager Alex Ferguson made 11 changes from the side that beat Chelsea in the Premier League on Sunday but they still had too much quality for hard-working Leeds.
"The first half was a tremendous performance by us, our movement off the ball and possession were very good," Ferguson told Sky Sports. "Leeds kept trying and trying (but) we just wanted to make sure we went through."
Asked about Owen he added: "His ratio of goals (to games) is unbelievable. His second goal in particular was fantastic... he's not getting the (number of) games he deserves."
Arsenal's terrible start to the season looked like continuing when they fell behind to James Collins's 16th-minute header for Shrewsbury but Kieran Gibbs levelled in the 33rd.
Alex Oxlade-Chamberlain, 18, put last season's runners-up ahead before the hour and Yossi Benayoun also got his first Arsenal goal to ease the pressure on manager Arsene Wenger.
"It was a bit nervy because we played against a good side," Wenger said on the club's website (www.arsenal.com). "I said before the game the lower divisions have improved in the last five years and you could see that."
Tottenham's campaign ended when 18-year-old substitute Massimo Luongo's penalty was saved by Stoke's Thomas Sorensen while Newcastle progressed when Fabricio Coloccini grabbed a last-gasp winner at Championship team Forest.
In other ties, Wolverhampton Wanderers beat second-tier Millwall 5-0 and Blackburn Rovers battled past stubborn third-tier Leyton Orient 3-2.
