Nathan Dyer broke the deadlock for the Swans before being withdrawn in the first half with a suspected fractured ankle as the Canaries clung on for a point at Carrow Road.
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 The winger was then withdrawn in the first half with a suspected fractured ankle.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 leveling proceedings in first-half injury time through Hooper's magnificent effort.
The hosts looked the more likely to find a winner in the second half, with centre-half Michael Turner coming closest when his header rebounded off the crossbar.
But neither side could fine the breakthrough as the shares were spoiled, with the point moving Norwich to 14th, while Swansea remained in 10th.
Vorm returned in goal after missing the 1-1 draw with Hull City because of a knee injury, while Norwich was unchanged from the side that beat West Brom 2-0.
Swansea enjoyed the better of the opening exchanges and were rewarded with the opening goal early on when Ashley Williams played a superb long ball that Norwich failed to deal with as Dyer ghosted in and confidently beat Ruddy.
The hosts could have leveled 10 minutes before the break as great tenacity from Hooper saw the striker outmuscle Williams, but he fired wide after Vorm reacted quickly to narrow the angle.
The game came to a temporary halt when Dyer was taken off the field on a stretcher in a challenge with Sebastien Bassong, with a potentially lengthy spell on the sidelines awaiting the 26-year-old.
With six minutes of time added on, Norwich pushed for an equalizer and Williams had to divert off the line when the ball ricocheted off Jordi Amat.
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 agonizingly 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.