From here, Oaks boss Ric Blank "thought we'd go and settle the issue, but we overplayed and let Woodford back into it on the break."
Oakwood had a warning when Woodford hit the post but a few minutes later they equalised, with Mitch Hahn finishing off a quick counter-attack.
The Oaks responded well, though, and restored their advantage, with Josh Bentley burying one from the edge of the box.
Oakwood "still didn't move the ball quick enough" for Blank's liking, but Bobbie Cousins and Josh Rose came off the bench and made a difference, as they ended up winning comfortably.
Wine converted another long-range special and Guy Helman slipped one in for number four at the death.
Oakwood joint-manager Ric Blank told JC Sport: "It was a good performance against a committed Woodford team but in the end we had too much.
"The weather in the first was a real leveller - it was shocking conditions. Stood on the sides the wind and rain were brutal and I can only think they affected both sides in an opening 10 minutes that lacked fluency.
"Overall a decent performance but now we have some real tests. Raiders, Southgate and Faithfold - that's our next three games."