Lukaku became the first Everton player to score in seven successive games in 40 years, since Bob Latchford in December 1975, when heading in Gerard Deulofeu's cross after 15 minutes.
The Belgium striker will bid to close in on Jamie Vardy's record of scoring in 11 straight games against Vardy's club Leicester next Saturday.
He has 28 Everton goals this calendar year, but might have had three more in a first half the visitors dominated to such an extent they could have been coasting at half-time.
Fit-again Leighton Baines hit a post and Arouna Kone was twice denied by Declan Rudd, yet Norwich responded little more than a minute after the restart through Wes Hoolahan.
Cameron Jerome might have won it for the Canaries, but fired over from Hoolahan's flick.
Norwich gave the ball away too freely in the first half and the best of their few forays forward came when Nathan Redmond sliced wide after running on to Gary O'Neil's fine pass.
The 2,600 visiting fans - wearing blue santa hats, with one of them in his swimming trunks - had a prime first-half view of Deulofeu's trickery on the right wing.
Lukaku dragged a shot wide across goal before Barcelona youth team graduate Deulofeu, who sparkled at times, laid on the opener.
Baines' corner was half cleared by Martin Olsson and O'Neil, allowing the Spaniard to cross for Lukaku to head in at the far post.
Lukaku tried to lay on a second, but Rudd was out quickly to thwart Kone, who soon afterwards was again denied in similar circumstances.
Another cross from the Everton right, this time from Tom Cleverley, could also have resulted in a goal for Lukaku but Russell Martin managed to get a touch to put him off.
Next Seamus Coleman crossed from the same flank
Lukaku swiped at mid-air but Baines was behind him, settled himself and rifled a shot against the post.
With Sebastien Bassong backing off, Lukaku made space for a shot which he dragged wide and he did so again before the interval.
The visitors' one moment of alarm came when John Stones landed awkwardly in challenging at Norwich's near post, but the England defender continued in the second half.
The muted half-time boos from the home faithful turned to delighted cheers when Ryan Bennett, on for Andre Wisdom, headed Robbie Brady's corner goalwards and Hoolahan forced it in despite the presence of Ross Barkley on the post
It was not pretty, but the Canaries cared not a jot as it was just their fourth goal in seven Premier League games.
Norwich's need for a goal was highlighted by their supporters serenading Everton substitute Steven Naismith as he warmed up
A bid from Norwich was rejected for the Scot in the summer.
And it was magnified further when Martin and Hoolahan combined on the right to craft a chance for Jerome, who could only lift the ball over when it appeared easier to score.
TWEET OF THE MATCH
"Take a point after the 1st half, much better 2nd and should've scored but Everton should've been out of site (sic)! Well sorted by the Gaffer" - Norwich boss Alex Neil earned former forward Darren Huckerby's praise.
PLAYER RATINGS
Norwich
Declan Rudd: 7 (out of 10)
Andre Wisdom: 4
Martin Olsson: 5
Russell Martin: 7
Sebastien Bassong: 5
Alex Tettey: 6
Nathan Redmond: 6
Gary O'Neil: 6
Cameron Jerome: 5
Wes Hoolahan: 7
Robbie Brady: 6
Substitutes:
Ryan Bennett (for Wisdom, 45): 6
Jonny Howson (Hoolahan, 84): 5
Everton
Tim Howard: 6
Seamus Coleman: 6
Leighton Baines: 7
John Stones: 7
Ramiro Funes Mori: 6
Gareth Barry: 6
Gerard Deulofeu: 7
Tom Cleverley: 6
Arouna Kone: 6
Romelu Lukaku: 7
Ross Barkley: 6
Substitutes:
Kevin Mirallas (Deulofeu, 69): 5
Darron Gibson (Barkely, 76): 5
STAR PLAYER
Lukaku
The big Belgian scored for a seventh straight game, but could, and should, have had more in the first half.
MOMENT OF THE MATCH
Fine work by Martin and a Hoolahan flick went unrewarded as Jerome's composure deserted him
The striker lifted the ball over from close range, when a goal would have put Norwich 2-1 up with 21 minutes to play.
VIEW FROM THE BENCH
Norwich made four changes following the loss at Watford, with Martin, O'Neil, Hoolahan and Jerome coming in, while there was good news for Everton and England, with Baines making his first start of the season, in place of Brendan Galloway
Wisdom was replaced by Bennett at half-time and Norwich seemed more defensively sound
Deulofeu went off with 21 minutes to go, but neither side could craft a winner.
MOAN OF THE MATCH
It was the proverbial game of two halves
What happened to Norwich in the first? Everton looked to be cruising
Had the visitors started the way they did in the second half, the Canaries could have claimed a crucial win.
WHO'S UP NEXT
Manchester United v Norwich (Barclays Premier League, Saturday, December 19)
Everton v Leicester (Barclays Premier League, Saturday, December 19)
Source : PA
Source: PA