John Ruddy dealt unconvincingly with a free-kick from Aguero, who then shot narrowly wide across goal.
The Argentina striker next ran at the Norwich defence and Ruddy made a sharp save to his right.
Norwich were set up not to concede, having last claimed a clean sheet on January 2. A 3-0 FA Cup third-round loss to City at Carrow Road was the first fixture of that 11-game run, but the visitors showed nothing like the same cutting edge this time around.
On loan Chelsea striker Bamford came close to upsetting City on his first start.
Nicolas Otamendi allowed the ball to bounce and Bamford waited patiently for it to drop. Once it settled, he sweetly swung his left-foot and the result was a 25-yard volley which Joe Hart watched hit his crossbar.
That effort and a tackle by Jonny Howson on Aguero seemed to galvanise Norwich. Matt Jarvis, who switched to the right when Nathan Redmond limped off with what appeared to be a calf problem, next went close.