The former Celtic striker showed neat interplay with Johan Elmander before unleashing a stunning volley that left Swansea goalkeeper Michel Vorm with no chance, and extended Norwich's unbeaten home run in the league to four matches.
Swansea made a bright start and took the lead after 12 minutes when Nathan Dyer latched onto a long ball and calmly lifted his effort over John Ruddy. The winger was then withdrawn in the first half with a suspected fractured ankle.
Norwich finished the half brightly before levelling proceedings in first-half injury time through Hooper's magnificent effort.
Chris Hughton's side finally drew level deep into first-half stoppage time, when Steven Whittaker's punt forward was chested down by Elmander and Hooper's spectacular strike from 30 yards was too good for Vorm.
Norwich started the second half in a similar fashion to how they ended the first, and shortly after the break Bassong met Nathan Redmond's driven corner but headed harmlessly wide.
Turner came agonisingly close to netting the winner with a looping header in the 68th minute that bounced back off the bar, before Hooper collected Jonny Howson's pass but was denied by a fine stop from Vorm.
Substitute Wilfried Bony nearly stole a late win for Swansea, who also saw half-hearted claims for a penalty waved away following Bassong's accidental handball, as he forced Ruddy into a diving save to his left.
