A 3-1 win at home to Swansea has set up a tough quarter-final at Arsenal - who denied the Toffees another Merseyside derby at Anfield by beating the Reds 2-1.
Everton's last trophy was their FA Cup win of 1995 and after getting to the final in 2009, where they lost to Chelsea, the players are looking to go one better spurred on by manager Roberto Martinez who won the competition with Wigan last May.
"We want to go as far as we can in the cup," said Coleman.
"It is going to be a difficult draw but you have to beat them all at some stage if you want to win and there are some good teams still in it.
"I've been here five years now and I've not won anything so it would be nice to win and there are other lads who have been here longer who want to win it as well.
"It would lift the club as a whole. Winning an FA Cup is massive for any club and it would be as big for us and a big stepping stone for other things."
Everton will hope by the time their tie comes around on the weekend of March 8/9 Arsenal will be distracted by a Champions League second-leg clash with Bayern Munich a few days later.
A cup run could impact on the Toffees' attempts to keep in touch with the top four but Coleman believes their squad has enough depth now injuries are starting to ease.
"We have Rom (Romelu Lukaku) coming back from injury so we are starting to get stronger after a few injuries with Antolin Alcaraz nearly back as well," added the Republic of Ireland international.
"We have got a good squad and for the things we are looking for this season we need a big squad."
A goal on debut for on-loan Monaco striker Lacina Traore was a welcome boost and Coleman believes having another option up front will help.
"Lacina has taken a while to get in because his injury but he has been brilliant and is a great asset to have on the pitch - and I am sure that goal will do his confidence the world of good," he said.
"He did well and thankfully he got a goal to get him going."
Martinez is also pleased to have a squad back virtually back to full fitness.
"We are strong in numbers now," said the Spaniard.
"All of a sudden we had a couple of players back and Antolin Alcaraz could have been available to be in the squad so I am really excited.
"It is about having everyone ready and I am really pleased with the numbers we have and I feel anyone is capable of coming on and making a big contribution.
"That will be the key from now until the end of the season."
Swansea's caretaker manager Garry Monk is confident his side will bounce back from their first defeat in three matches since he took over from the sacked Michael Laudrup earlier this month.
"I never accept it (defeat), not without a fight. That is the way I have always been," said the former Swans defender, who now has to prepare for Thursday's Europa League tie at home to Napoli.
"The players are hurting, there are no bigger critics than themselves. I can live with it as long as we learn from it."
Defender Kyle Bartley was forced off with a hamstring injury in the first half but there is better news on other players who have been sidelined.
"Michu trained towards the end of last week; he was okay but will need some work to catch up. We will make a judgement call (for Napoli).
"Jonjo (Shelvey) should be back in training this week and the same with Alex (Pozuelo).
"We will assess Barts and then hopefully it will be a full squad."