The 17-year-old Liverpool striker drilled home the winner just four minutes after coming on as a second-half substitute to keep Wales unbeaten in qualifying and four points adrift of Group D leaders Serbia.
Chris Coleman’s side also cut the gap to second-placed Republic of Ireland, held 1-1 in Georgia, to two points with three matches to play as they attempt to reach the World Cup finals for the first time since 1958.
“The ball just came out of the air and I just tried to shoot it as quickly as I could,” Woodburn told Sky Sports, after becoming Wales’ second youngest goalscorer in history at 17 years and 322 days, behind only Gareth Bale.
“It’s a great feeling, I couldn’t ask for much more from the fans. I won’t sleep tonight.”
Bale returned from suspension at the Principality Stadium after missing the 1-1 stalemate with Serbia in June, the fifth time in sixth qualifiers Wales had been held to a draw.
Arsenal midfielder Aaron Ramsey fired a left-footed drive narrowly over inside the opening 10 minutes, while Marko Arnautovic curled over with just Wayne Hennessey to beat after a terrific through-ball from Julian Baumgartlinger.
Austria defender Martin Hinteregger bravely blocked a firmly-struck effort from Ramsey early in the second period before Heinz Lindner palmed behind a vicious, dipping shot from Bale.
Woodburn replaced Tom Lawrence on 69 minutes and Coleman’s gamble reaped immediate rewards as the youngster controlled a headed clearance before rifling beyond Lindner from 25 metres to spark jubilant celebrations.
Captain Ashley Williams diverted an Arnautovic attempt over his own crossbar as Austria searched desperately for an equaliser, but Wales nearly added to their advantage twice in stoppage time.
First, Bale’s acrobatic overhead effort was superbly clawed away by Lindner before Hal Robson-Kanu’s drive clipped the base of the post with virtually the final kick of the game.
“It was one of those games were it was to and fro. Austria will feel hard done by but on the balance of the second half we were the stronger team,” said Coleman.
“We have to do the same again in Moldova (on Tuesday). We will be okay if we do what we did in the second half tonight.”
Earlier, Ireland were frustrated in Tbilisi as Valeri Kazaishvili cancelled out Shane Duffy’s fourth-minute goal for the visitors.
Serbia held onto top spot in the group thanks to a 3-0 home win over Moldova, with Mijat Gacinovic, Aleksandar Kolarov and Aleksandar Mitrovic getting the goals.
Martin O’Neill’s Ireland dropped two points behind Serbia, who they host in Dublin on Tuesday, while Austria’s prospects now appear all but doomed ahead of the visit of Georgia.