Debutant Eddie Johnson fired over for the hosts but City raced away to score the opener on five minutes when Andy Keogh tucked in Steven Caulker's pass.
Graham wasted a great chance in injury time when he hooked a shot wide, and then goalkeeper John Ruddy came to Norwich's rescue to tip over Steven Caulker's bullet header in a breathless finish.
With the snow still teeming down in the second half Tottenham cranked up the pressure, though they were fortunate not to concede a penalty when StevenCaulker looked to have tripped substitute Wayne Rooney inside the area.