The Canaries found themselves behind on two occasions after Nikica Jelavic netted twice for the Toffees but a first Norwich goal from Jonny Howson and a lat finish from the ever-reliable Grant Holt allowed the home side to grab a point.
Croatian ace Jelavic kicked things off in the 22nd minute when he neatly flicked home Phil Jagielka's cross from the right side of the area with the outside of his boot to continue his golden start to life on Merseyside.
But Norwich struck back 17 minutes later when Howson slotted home from close range from a superb ball from Wes Hoolahan to get the home fans up and about.
That bubble of hope, however, was burst when Jelavic remarkably became Everton's joint top scorer in the league with his fourth goal in his eighth appearance when he expertly tucked home Steven Pienaar's low cross from the left to again give the visiting Toffees the ascendency.
In a game of ebbs and flows, Norwich again pulled the lead back just under 15 minutes from time when skipper Holt produced a cool finish for his 15th of the campaign after some good work from substitute Aaron Wilbraham to put Paul Lambert's side on 40 points in their first season back in the top flight which will surely see them to safety.
For David Moyes' men, it was an opportunity missed to score a big away win but nevertheless thy remain in the top half of the table with six matches left.
TRIBAL ASKS: If Everton had Jelavic all season, would it be safe to say they may be pushing for European football again?