Craig Bellamy opened up the account in injury time of the first half with his first goal of his second coming at Anfield against his old club when he took possession of the ball on the edge of the area and with a little help from Marc Tierney the ball was in the back of the net at the Kop End.
Not to be outdone, Norwich's Grant Holt came off the bench to score the leveler when Anthony Pilkington supplied a very dangerous ball into the area and the former lower level player attacked it with some vigour to head home past Pepe Reina.
Liverpool were unlucky in hitting the woodwork on three occasions and having Luis Suarez's injury-time volley brilliantly saved by Canaries keeper John Ruddy but the Paul Lambert's men were resilient enough to take a point from an extremely tricky away trip.
Martin Skrtel's third-minute header from Charlie Adam's corner beat Ruddy but not the post before the Norwich gloveman pushed a Suarez attempt, from Bellamy's cheeky run and cross from the left, onto the upright.
Six minutes into the second half, Suarez was again unlucky when he was through in the area but his flick with the outside of his foot came off Norwich defender Russell Martin and onto the post for the third time.
Both sides had good chances late, notably Holt's second opportunity to score a header from a Pilkington cross but Reina was on hand to swat away before Reds skipper Steven Gerrard put it on a plate for substitute Andy Carroll but the big man put his header wide when on most other days he would have tucked it away without question.
Moments later, Ruddy produced a stellar save from Suarez' pointblank volley to give his side a share of the points in a very open contest which sees Liverpool remain fifth whilst Norwich are just three points behind in seventh.
TRIBAL ASKS: A draw with Man Utd one week and then with Norwich at home the next. Does Dalglish have a consistency problem or are the Canaries actually a decent side?
Yellow cards:LIV - nil NOR - Marc Tierney 34'