The Liverpool youngster found a way through from range soon after coming off the bench to settle a cagey affair in the Welsh capital.
After edging the first half, the visiting side struggled to create any serious opening in the second 45 and were eventually undone by Woodburn.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Hennessey; Gunter, Chester, Williams, Richards, Davies; Lawrence, Ramsey, Bale, Edwards; Vokes
SUBS: Ward, A. Davies, King, Robson-Kanu, Evans, Ampadu, Ledley, Lockyer, Collins, J Williams, Woodburn, Watkins
STARTING XI: Lidner, Lainer, Dragovic, Hinteregger, Prodl, Ilsanker, Arnautovic, Alaba, Sabitzer, Harnik, Baumgartlinger
SUBS: Bachmann, Kuster, Wober, Schaub, Bauer, Grillitsch, Kainz, Danso, Heirlander, Gregoritsch, Janko, Laimer
Chris Coleman: "I don't think either team is going into the game looking for a draw. Both teams are capable of scoring goals with good, exciting players. I think it will be open and a great game. A draw doesn't do either of us any good so you would imagine something will have to give. This was always going to be a tight group, the teams are very similar and four teams are very strong. I feel it will go right to the wire."
Ashley Williams: "It's one we feel we must win. That's what we haven't had, that momentum in that campaign, which has been frustrating. But I do feel like it changed after the Serbia game [in June], which was still a draw but we felt like it was a good draw for us. It felt like the start then. If we can win this game, then we can keep this momentum going for the last three after this. That will be the plan and pick up a bit of steam to finish the campaign."
WALES SUBS: Ward, A. Davies, King, Robson-Kanu, Evans, Ampadu, Ledley, Lockyer, Collins, J Williams, Woodburn, Watkins
AUSTRIA SUBS: Bachmann, Kuster, Wober, Schaub, Bauer, Grillitsch, Kainz, Heirlander, Gregoritsch, Janko, Laimer