Defender Oluwo powered home from a corner to se cure a deserved victory for Dean Brennan's men against an injury ravaged Wood.
"We dominated the game and I'm delighted with the result but, we need to create more chances if we're going to have a really successful season," Brennan beamed.
"There were some very good individual performances today from my players. We showed great character."
Barnet controlled a lot of early possession and fashioned the first chance of note, although Harry Pritchard's free-kick from a good position could only hit the wall.
Two minutes later, Wood had a good chance to take the lead when a delightful Kwesi Appiah pass picked out Lee Ndlovu but he was off target with his effort.
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