The Canaries dominated the game right from the start and never looked like losing once Jordan Hugill scored just his second goal of the season in the 36th minute.
Bristol City failed to have any real impact in the game as they created just three chances and they struggled to deal with the lively Emi Buendia and Todd Cantwell.
• Story of the match
As expected, Norwich dominated the early stages of the game with Bristol City struggling to get hold of any meaningful possession.
Daniel Farke's men looked dangerous right from the off as Cantwell had an effort saved just two minutes in which laid out the home sides intentions.
As Norwich continued to camp in the Bristol City half, they began creating more goal scoring opportunities.
After 20 minutes the in-demand Buendia had a shot blocked by Robins defender Tomas Kalas as it looked to be heading in.
Bristol City goalkeeper, Daniel Bentley then pulled off a succession of excellent saves to deny Scotsman Kenny McLean and front man Hugill
Norwich's pressure finally paid with 36 minutes on the clock. The ball was well recycled by the Canaries before Cantwell whipped in a driven cross into a dangerous area and Hugill made no mistake in heading past Bentley for his second goal of the season.
The half flattened off after the goal and Bristol City failed to have much impact in the first 45, having no shots on target and just two touches in the Norwich 18-yard box.
Dean Holden's Bristol City did however begin the second half with much more intent than the first. They had a much more even share of possession and looked more threatening in and around the Norwich penalty area.
Norwich weathered the storm, defending resolutely and they began growing into the second half and dominating like they did in the first.
Buendia came close in the 63rd minute when his left footed drive just whistled past the post and Max Aarons had a deflected shot parried away by Bentley.
The Canaries pressure again was rewarded with a goal in the 76th minute when Aarons twisted and turned past Nahki Wells before driving in a powerful low cross which ricocheted off a Bristol defender and on to Hugill's chest as he calmly directed the ball into the net.
Some good game management from Norwich saw them comfortably see out what looked to be a much tougher affair on paper.
That win puts Norwich seven points clear at the top of the Championship with Barnsley up next in the FA Cup.
• Takeaways from the match
Norwich look destined for the Premier League
After struggling so much last season, Norwich fans will be pleased to finally see their team winning again.
Tonight was their third successive win in the league and the three points put them seven points clear of second place Swansea City.
Bristol City is not an easy game but tonight, Norwich made it look easy. Their composure in possession made it very difficult for Bristol City to get hold of the ball.
The two Norwich full-backs, Jacob Lungi Sorensen and Aarons were constantly bombing on and they forced Bristol City into their own half, not letting them out.
Norwich were constantly a threat down the flanks, helped by the great play of their right and left back.
Farke's men always looked like they were going to score or cause danger and they never looked to break a sweat when defending.
The way they were able to ease past a strong Bristol City side portrays, how, if they carry on, there will only be one outcome at the end of the season and that will be promotion.
Bristol City need more quality going forward
In tonights game, the Robins struggled to have their own say on proceedings.
At times in the first half, they defended very well and it was only thanks to some brilliant saves from Bentley and heroic defending from Kalas that they were still in the game.
What they needed was an out ball. Someone higher up the pitch who would chase lost causes and bring others in to play.
They needed somebody to make something out of nothing and have an effect on a game that they were getting dominated in.
Bristol City need to offer more if they are to challenge for a playoff spot. There was no attacking spark tonight and if they want to start winning more games, they have to do more in the final third.
• Stand out player - Jordan Hugill
Not only were his two goals the difference between the two sides, his willingness to run in behind and come short always gave Norwich an option when they were under pressure.
Hugill held the ball up well and looked like scoring whenever a chance came his way. Even though there were many good performers tonight for the Canaries, Hugill was the main stand out.