The 17-year-old, who has been in and around the Liverpool first team of late, was introduced as a 68th-minute sub and netted soon after to settle a nervy contest at the Cardiff City Stadium.
Woodburn's goal lifts Wales to within two points of second-placed Republic of Ireland in Group D, while leaders Serbia are four points better off heading into the final three matches.
After dominating the opening 10 minutes without creating any openings, Wales were soon pushed back and were arguably fortunate not go into the break a goal down.
An Aaron Ramsey shot aside, which was always rising over the bar, it was the visiting side who looked the more likely to make a first-half breakthrough as Marko Arnautovic and David Alaba began to pull the strings.
The West Ham United attacker could only send the ball over the bar from a tight angle, however, and Marcel Sabitzer also sent a shot inches wide of the target.
Hennessey was called into action for the first time 41 minutes into the contest, getting down well to deny RB Leipzig forward Sabitzer after he turned and shot inside the box.
Austria, unbeaten in their last four heading into this match, were continuing to look dangerous in patches but they were no longer in complete control as Ramsey burst forward and brought a fingertip stop out of Lindner.
Coleman turned to Hal Robson-Kanu and debutant Woodburn for the final quarter of the match in a desperate attempt to keep his side's qualifying campaign alive, following a previous run of five successive draws.
It took all of five minutes for Woodburn to mark his debut in stunning style, picking out the back of the net from 23 yards out to give Wales the lead they craved.
Arnautovic so nearly spoiled the party when working some space and seemingly getting the beating of Hennessey, only for Ashley Williams to block the ball with his head.
Austria struggled to create any real chances in the remaining time, with Robson-Kanu and Bale both coming close to rounding things off in the dying embers - the former being denied by the post from the final meaningful act.