The Canaries booked their place in the fifth round with a 2-1 win at West Brom on Saturday, and will now host npower Championship side Leicester for a place in the quarter-finals.
However, Lambert insists the focus remains on securing top-flight football again, with his side currently ninth with 29 points.
"I can understand people wanting to stay in the Premier League first and foremost for the simple reason that it is the greatest league, and that is not to undermine the FA Cup at all," Lambert said in the Eastern Daily Press.
"If you asking me which I prefer, then the Premier League status is absolutely vital for this football club.
"The lads are buoyant but win, lose or draw you let it go and look at the next game.
"We have a really tough one [at Sunderland] on Wednesday night, but we will go up there in great form and with great momentum and do everything we can to win."
Lambert rotated his squad at The Hawthorns, where Norwich had also won 2-1 in the league a fortnight ago.
Wales striker Steve Morison, midfielders Bradley Johnson and Andrew Surman along with regular goalkeeper John Ruddy were all rested and will come back into contention for the trip to the north-east.
"We have got another game on Wednesday that is another huge, huge game for the football club so we changed a few of them," Lambert said.
"We never played Morison, Surman or Johnson nor Ruddy. I just wanted to give those lads a break because they have played a lot of games."
While Norwich fans may already be weighing up potential FA Cup quarter-final opponents, Lambert insists they cannot take the challenge of the Foxes lightly next month.
"We played them twice last year in two really tough games, although it is a new manager now. It will be a really tough game, but it is one we will look to go and win," Lambert added on www.canaries.co.uk.
"It is a home tie and I think that is all you ever look for in a cup draw, regardless of what the opposition was going to be."
