The Championship side needed a replay to secure their date with the Reds at Anfield, winning 3-1 in the third-round replay at the Memorial Stadium.
The result confirms that Liverpool will take on Norwich at home on Sunday, 28 January at 14:30 GMT for a place in the last 32.
The game against Norwich at Anfield will be televised in the UK, with ITV selecting the fixture for broadcast.
Reunion for Klopp
Norwich, who are currently 11th in the Championship, are managed by David Wagner – a former teammate of Jurgen Klopp and the best man at the Liverpool boss' wedding in 2005.
Speaking on the upcoming 'reunion' with his old friend, after beating Bristol Rovers, Wagner told reporters: “I haven’t checked my messages yet but I wouldn’t be surprised if Jurgen has already been in touch now we’re through.
“I haven’t seen Jurgen in person since his birthday party last year so it will be great to have a bit of reunion with him.
“But it’s not about us and above all it’s all about making sure my team go to Anfield and do Norwich City proud.
“A tie at Anfield is a really big prize for my team and I know my players will relish the opportunity to take on the Premier League's best players. But it’s not something that should hold any fear for my players and I will be telling them that we will go for it.
“It will be a great game but there are important Championship games between now and Liverpool, but I promise we will be ready and will give a good account of ourselves.”
Klopp has won seven out of seven games against Norwich in his time as Liverpool boss – including a 2-1 win in the FA Cup on the way to winning the trophy in 2022.
In three previous contests against his old friend Wagner, Klopp has won all three.