Fulham run out 3-1 winners.
There were positives for John McGreal's side though, as Owen Garvan, Kane Vincent-Young and James Bransgrove continued their recoveries from injury.
Possession was lost in midfield and Woodrow coolly stroked his effort into the bottom corner to put his side through to the next round, where they'll face West Ham.
The U's worked hard for an equaliser but the more experienced Fulham side always looked dangerous and made the game safe with a couple of minutes to go.
The equilibrium was short lived though, with Dean O'Halloran scoring at the near post from a right wing corner.
Seven minutes after the break, it was all square as Chris Regis' angled shot was deflected past Rodak to make it 1-1.
Alexander Kakaniclic scored the only goal of the first half, smashing a shot off the underside of the bar and behind the line.
Fielding a much younger eleven than Fulham, the U's worked hard but were under a lot of pressure throughout proceedings.
It was a win that the Londoners deserved, with James Bransgrove the busier of the two keepers. The U's have been beaten 3-1 by Fulham in the U21s Premier League Cup.