Leon Osman's deft second-half finish rescued a point for Everton as they finally broke Norwich's resistance.
The hosts dominated throughout and created a series of chances, but it was City who took the lead when Grant Holt turned and shot low into the corner.
Louis Saha came closest to levelling before the break with a fizzing low effort and the chances kept coming.
The pressure finally told when Osman diverted in Royston Drenthe's strike to deny the Canaries a rare away victory.
Substitute Drenthe then almost won it with a fierce, swerving strike that overworked visiting goalkeeper John Ruddy did well to block with his shoulder.
Before Drenthe's late cameo, Norwich looked set to secure their second away Premier League victory this season thanks to a combination of resolute defending and sloppy home finishing.
But Everton's determination and refusal to accept a fifth home defeat in eight league games this season finally told.
As commendable as Norwich's display was, Everton could easily have wrapped up the game by the time Holt had fired the Canaries ahead.
Osman, Marouane Fellaini and Phil Neville got to grips with the midfield and allowed Everton to dominate possession and territory from the outset.
Fellaini had the first chance after a delightful first-time pass from Tim Cahill, but the big-haired Belgian's poked effort was brilliantly blocked by Ruddy.
Osman then cut in from the left and shot towards goal, but the effort was headed away to safety by the backpedalling Russell Martin.
Magaye Gueye also volleyed wide after Ruddy flapped at a clearance following a series of corners and Cahill sent a tame effort wide.
A home goal seemed inevitable, but when the deadlock was broken it was the visitors celebrating.
Canaries forward Steve Morison knocked David Fox's floated free-kick back into the danger area and Holt showed great strength, composure and awareness to swivel neatly in the six-yard box and shoot into the net via the inside of the far post.
Holt's contribution - his seventh goal in 14 league games this season - could have ended soon after when he appeared to elbow Fellaini in the jaw.
The Belgian clumsily challenged Holt and seemed to catch him around the face. The burly striker reacted before instantly protesting his innocence.
Referee Lee Probert decided not to book either, but had he got a good view of the incident, Holt could easily have been dismissed.
With Everton rattled, Wes Hoolahan squandered a great chance to make it 2-0 when he profited from a Neville slip, but overhit his cross.
The same pattern of relentless home pressure continued after the interval, with Saha the biggest threat.
DID YOU KNOW?
Norwich's last top-flight away win over Everton was also their biggest Premier League victory on their travels. Efan Ekoku scored four goals as the Canaries coasted to a 5-1 success.
Osman had a shot turned around the post by Ruddy after a great run and unselfish pass from Saha and the Frenchman then forced a brilliant instinctive block from Ruddy following Denis Stracqualursi's flick-on.
Martin also blocked Saha's shot from the edge of the box after a rare, rash Ruddy moment saw the keeper come racing out of his goal.
At the other end, Norwich's Holt almost headed a second.
But the Goodison Park gloom was lifted when Drenthe's low strike was cleverly flicked in by fit-again midfielder Osman.
The home pressure increased even more after the leveller. Saha went close again and Drenthe twice had efforts saved by Ruddy.
Marc Tierney's stunning tackle also denied Stracqualursi as Everton were forced to settle from a point - only their eighth from eight home games this season.
Everton manager David Moyes:
"I'm delighted we got a point in the end but disappointed we didn't win. If we had scored a goal in the first 20 minutes I think we would have gone on and got the win.
"We started very well had some good opportunities but didn't take them.
"Overall we played okay. We lost our way a little bit before half-time but we kept going and we deserved a win.
"But a draw was more than I was thinking I was going to get today towards the end."
Norwich City manager Paul Lambert:
Lambert pleased with a point
"This is a really hard place to come and we battled really strongly and defended really strongly when we had to but also looked a threat.
"They came rolling right at us and you weather the storm, that is what you have to do and that is what we did at Anfield.
"Any point away from home is good. Holt has been great for the last two-and-a-half years I have been here. I think he has scored 50-odd goals in 100 or so games and in anyone's book that is a phenomenal ratio.
"He has always been a handful and it is a great testament to him he is probably getting the reward from it. It was a brilliant finish."
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