Grant Holt and Simeon Jackson fired the Barclays Premier League side 2-0 ahead inside 12 minutes. However.
Jay Rodriguez's looping header reduced the deficit and then City skipper Holt missed a penalty.
The Canaries were soon back in command when Andrew Surman swept home on the hour and substitute Steve Morison tapped in number four from close range.
Lambert believes the result was testament to the progress the group have made since winning promotion last season.
"To go through is the most important thing, but we dominated the game from start to finish," said Lambert, whose side are currently ninth in the Premier League table.
"Burnley have been doing well in the Championship, but hopefully we have moved on from last year, when we had two really tough games against them.
"The Premier League does that to teams, with the level and the intensity you play at. It makes everybody step up to the plate and it was important nobody went below that."
Lambert added: "All you can ask for in the next round is a home draw, but no matter who you will get, it is going to be extremely hard."
Norwich were awarded a penalty on 31 minutes when Andrew Croft's cross struck Charlie Austin on the arm.
However, Holt - who had never missed a penalty since joining Norwich in the summer of 2009 - drilled his shot wide of the left hand post.
Lambert, though, was not about to lambast his talisman striker, who was the leading scorer in both of their successive promotion campaigns coming up from npower League One.
"He has scored so many vital ones for us and I am sure he will do so again," the Norwich manager said.
"The performance of the team overall was very good and Grant scored a really good goal."
Lambert had resisted the temptation to make wholesale changes to his staring XI, retaining the back four from the 2-1 win at Queens Park Rangers.
Reserve keeper Declan Rudd was given a chance to impress, with number one John Ruddy absent after his partner gave birth last night.
Rudd had to make two early saves as Clarets made a bright start with Ross Wallace and then Rodriguez shooting from distance.
Burnley manager Eddie Howe had seen his build-up hampered by the suspension of Kieran Trippier while fellow defender Michael Duff missed out through injury.
Midfield duo Marvin Bartley and Keith Treacy had both been laid low with illness and could only make the bench.
Howe said: "Norwich have definitely improved and got some good players with consistent results, as such they are hard to beat.
"We were not at full strength today and when you come to a Premier League ground, you need to have your best team with you to have any chance.
"We will learn from it and hopefully have a good second half to the season."