After commanding much of the game Everton found themselves a goal down at Carrow Road through an unlikely Steven Whittaker strike early in the second half.
Two goals in four minutes just after the hour - the first a stunning effort from Ross Barkley before Seamus Coleman converted a tap-in - looked like putting Roberto Martinez on course for a win in his first game in charge of the Toffees.
But Norwich's club-record signing Van Wolfswinkel produced a fantastic header with a little under 20 minutes remaining to earn his new side a point.
The home side brought the game to life after 15 minutes when Bradley Johnson's pass was knocked into the path of Elliott Bennett by Wes Hoolahan, but £8.5million summer signing Van Wolfswinkel's header was deflected behind for a corner.
Van Wolfswinkel was involved again moments later as he latched on to a fine long ball by Russell Martin and crossed towards Hoolahan, only for Sylvain Distin to make a diving clearance in front of the Republic of Ireland international.
That period of domination for Norwich seemed to stir Everton and they again took control of proceedings with Kevin Mirallas firing just wide of John Ruddy's right-hand post after a neat interchange with Nikica Jelavic.
The Belgium international was on the end of a free-flowing move by Everton early in the second half which saw him thrash an effort in to the side netting following good work down the right by Leon Osman and Coleman.
But it was Norwich who took the lead shortly afterwards courtesy of right-back Whittaker.
The Scotland international stumbled his way through a couple of Everton challenges before opening up his body and curling a left-footed effort past Tim Howard and against the post.
Whittaker then reacted quicker than any of the visitors' defenders and was on hand to roll the rebound in to the empty net and give the Canaries the lead just six minutes after the interval.
Everton responded well to falling behind and equalised in superb style just 10 minutes later as Barkley hammered in his first goal for the club, arrowing an effort past Ruddy and deservedly levelling for the visitors.
It did not take Everton long to establish the lead their play for much of the game had warranted.
Steven Pienaar slipped a smart ball in to Jelavic, whose shot was well saved by Ruddy, but Coleman was following up and had the simple task of rolling home the rebound to turn the game on its head.
Everton, however, had even less time to enjoy their lead than Norwich had done earlier in the game as the hosts refuse to lay down, pulling level after 71 minutes.
Whittaker was again heavily involved as his speculative left-foot strike from the edge of the area turned into the perfect cross for Van Wolfswinkel to direct a header past Howard and level the match once more.
Visiting substitute Steven Naismith forced a smart save out of Ruddy from close range with Martin making a superb block from Jelavic on the follow-up and Coleman saw a late drive fly just wide as Everton did more to find a winning goal that never came.