Woods, 21, upgraded the bronze he won at the recent X Games in the slopestyle discipline, which makes its Olympic debut in Sochi, as he took second behind the USA's Thomas Wallisch.
It's Great Britain's first world skiing medal since Richard Cobbing's aerials silver in Altenmarkt 20 years ago.
Woods, who has also posted a succession of World Cup podiums this season, must have fancied his medal chances after making the final as second qualifier and finished just 2.2. points off top spot, while American Nicholas Goepper.
Meanwhile, British team-mate Katie Summerhayes just missed out on bronze in the women's slopesyle.
Summerhayes, 17, improved her qualifying position by one place to finish fourth as Canada's Kaya Turski took gold ahead of team-mate Dara Howell, while USA's Grete Eliassen claimed bronze.