The Canaries had to come from behind at home, however, after Nathan Dyer's opener after 12 minutes, lofting the ball over the oncoming John Ruddy after capitalising on Sebastien Bassong's mistake.
Dyer suffered a game-ending injury on 40 minutes, however, after buckling under the challenge of Bassong with what appeared to be a fractured ankle.
That gave Norwich the impetus to chase an equaliser and, after a period of dominance, it was Hooper's stunning volley deep into injury-time which grabbed the equaliser and, after an entertaining but goalless second half, a deserved point at home.
Swansea's passing style paid dividends early on as they settled a little quicker than their hosts, with Pablo Hernandez's snapshot providing the first glimpse of goal for either side.
Jonathan de Guzman will have hoped to do better with a 25-yard free-kick on 10 minutes, but moments after Swansea were ahead.
An ambitious long ball from Ashley Williams left Dyer and Bassong challenging for a header, with the diminutive winger somehow coming out on top.
Dyer was left to chase onto his own header, and the forward's lightning quick pace helped him reach the ball before the rushing Ruddy, giving Dyer the easiest of finishes as he helped it over the keeper.
Norwich battled for an equaliser afterwards, and came close on 35 minutes when Hooper robbed Williams of the ball before hitting his shot high and wide when well-placed.
Five minutes later, Dyer was stretchered off following his and Bassong's tussle, and Norwich were able to capitalise not long after.
Bassong's clearance found Johan Elmander on the edge of the area, and the Swede's knockdown was perfectly weighted for Hooper to chest the ball up and volley home past visiting keeper Michel Vorm from fully 25 yards for a stunning equaliser deep into injury-time.
Nathan Redmond nearly put his side ahead just one minute later, twisting and turning from the right wing but flashing his shot wide.
The second half was just as open as the first, as Bassong, Steven Whittaker, Hernandez and Williams all traded early opportunities, but to no avail.
Dyer's first-half replacement, Alejandro Pozuelo, then played a delightful one-two with Hernandez before blazing over on the hour mark, with both sides chasing a winner.
On 68 minutes, a poorly defended corner gave Michael Turner the chance to head the hosts ahead, but his powerful effort cracked back off the crossbar before rebounding out off Jordi Amat's head.
Norwich continued to press as first Redmond then Hooper forced excellent saves from Vorm, but Swansea nearly stole it at the death when substitute Wilfried Bony curled wide with Jonjo Shelvey well placed.
The sides eventually had to settle for a draw, however, pushing them both further towards mid-table.