The Magpies open their campaign on Saturday with Blackburn's visit to Tyneside in the third round with the city still revelling in Wednesday night's heroics against Premier League champions Manchester United.
A 3-0 win over Sir Alex Ferguson's men which keeps them in touch with the top six sent spirits soaring after a run of just one win in eight games.
It also marked the final involvement, for a few weeks at least, for in-form striker Demba Ba and key midfielder Cheick Tiote, who are now heading off to join their Senegal and Ivory Coast team-mates respectively for the African Nations Cup.
However, Gutierrez is confident that the squad is strong enough to ensure their interest in world football's oldest cup competition is preserved for when the pair return.
He said: "We have the players. We don't want to throw up the cup, we want to focus on that tournament as well because it is a beautiful competition.
"If you have good draws, you can maybe reach the semi-finals or even the final.
"First, we must think about the first game, but of course, we want to keep going in the competition."
The sides have met twice already this season with Newcastle winning 3-1 in the league at home on September 24 and Rovers turning the tables in the Carling Cup fourth-round encounter the following month, edging an extra-time thriller by the odd goal in seven.
Gutierrez said: "That was a crazy game, but now we have the chance to get our revenge and continue in the cup."
Manager Alan Pardew, who knows the cup competitions represent the Magpies' best chance of ending their 42-year wait for silverware, was devastated to have passed up that opportunity, and is in no mood to allow a repeat.
However, he is conscious that the gruelling holiday schedule and the loss of Ba and Tiote could have an impact.
Pardew said: "It's important because if you look at the competition, you would have to put us down as seventh favourite because that's where we are in the league, and the top two are playing each other, so that gives us a chance to win a trophy.
"We have to attack it and that's what we are going to do. It's very close to my heart. I have had some fantastic experiences in the cup.
"But this turnaround is not helpful for us. We had a similar turnaround last year when we lost to Stevenage.
"It's very important that we get ourselves as fresh as we can be and make sure we are focused on the game because Blackburn are going to make sure it's not going to be an easy game for us.
"They have had four or five days' rest and they know the effort we put in the other night, so I expect them to come at us.
"It's only the third round. It's the first game for us, so it's early days, but you mustn't take your eye off the prize.
"To go to Wembley and to lead this club out would be a dream for me, and that's what we hope will happen."
Striker Leon Best and French midfielder Hatem Ben Arfa seem likely to be the main beneficiaries of the exodus to Africa, although Pardew has warned his defenders, who were so resilient against United, may have to be equally pugnacious once again on the evidence of the Carling Cup tie.
He said: "Going on that showing, there's not going to be a lot between the teams.
"They are fantastic at set-plays. They work on them and we are going to have to be on our guard."
