The home side opened the scoring during the early stages of the second half as Steven Whittaker buried his own rebound before Everton looked to have won it when they scored twice in quick succession through Ross Barkley and Seamus Coleman.
However, a debut goal from Ricky van Wolfswinkel with just under 20 minutes remaining allowed Norwich to press in the closing stages as the two sides ultimately shared the points.
Norwich boss Chris Hughton was able to include new signings Van Wolfswinkel and Nathan Redmond in his starting line-up, but Leroy Fer’s debut will have to wait due to suspension and Robert Snodgrass missed out with a groin strain.
None of Everton’s transfer window acquisitions made it into the starting XI, as Gerard Deulofeu, Joel Robles and Arouna Kone all started on the bench, while reported Manchester United target Marouane Fellaini appeared from the beginning.
Everton began as the stronger side but Norwich did grow into the match, and only the stretched right leg of Sylvain Distin prevented a clear-cut chance in the 17th minute, as Wes Hoolahan met Van Wolfswinkel’s cut-back from the left, but the hosts had to settle for a corner.
Neither team managed to stamp their dominance on the match in the first half, but Everton did register two good efforts close to the half-hour mark. Kevin Mirallas shot just wide from outside of the area and then Fellaini fired well over from a tight angle to Ruddy’s left.
After going in level, Norwich came out after the break with renewed determination, and it took them just seven minutes to take the lead.
Right-back Whittaker broke forward down the right and managed to hold off the challenges of three Everton defenders before curling a left-footed shot towards goal. His effort came agonisingly back off the far post, but the Scotland international was alert enough chase down the rebound and knock it home.
The lead lasted less than 10 minutes, though, as Everton responded in the best possible fashion.
Coleman was the architect when he cut in from the right, skipped past two tackles and laid the ball back to the edge of the area for Barkley, who hammered a left-footed drive past Ruddy to level things up.
After 65 minutes, Everton thought they had completed the turnaround, as Nikica Jelavic broke into the left edge of the penalty area and fired a shot across goal, where Coleman was on hand to tap in after Ruddy’s initial save.
Norwich, though, had other ideas.
With 19 minutes left Whittaker sliced a left-footed volley into the danger area from the right and Van Wolfswinkel rose above Phil Jagielka to head the ball into the top corner to level things up once again.
Everton had one final chance in the closing moments when Norwich failed to clear Steven Pienaar’s left-wing cross, but Ruddy impressively stopped Steven Naismith’s close-range effort before Russell Martin blocked from Jelavic.
