Carter, who has amassed a record 1250 points during his 85-cap Test career, tore a groin tendon in kicking practice last Saturday, ending his participation in the tournament and leaving a nation wondering whether their 24-year wait to lift the William Webb Ellis trophy can be realised without the inspirational playmaker.
Slade assumed the number 10 jersey for Sunday's defeat of Canada and Carter sent his team-mate a text message "encouraging him to make the most of this opportunity and enjoy it".
Carter believes the All Blacks can cope with his absence, beginning with the quarter-final against Argentina on Sunday in Auckland.
He said: "The team are pretty focused. If we lose any player we've got enough depth and we've got such a good squad that players come in and fill their roles.
"We're in the play-off stages; we've got effectively a final this week. Anything can happen.
"We just have to go out there and put everything on the line like there's no tomorrow."
Carter revealed a contingency plan had been put in place in case of injury and All Blacks assistant coach Steve Hansen insisted the experience was part and parcel of sport.
Hansen said: "All sports people know that this is a possibility.
"There's a lot of sports people that have had their moment taken off them. Yes, it's gut-wrenching, but you move on.
"He's doing that the best he can. I'm sure he'll support the team the best he can.
"What the team needs to do and what Colin Slade needs to do... Colin first of all needs to understand that he's Colin Slade - he doesn't have to be Dan Carter.
"He just needs to be Colin Slade and what the rest of the team needs to do is do their job a little bit better and take up the slack that might've been left without him (Carter).
"Our job now is to get on and win this World Cup. We've got a quarter-final and that's where the focus is."
Hansen believes Carter will respond from the blow.
The assistant coach added: "There are disappointments in your career and adversity makes you a better person.
"I'm sure DC will come through this even stronger and an even better man than he is now - and he's a top bloke.
"He'll look forward to playing rugby again. And one of the good things that might come out of it is he might play in the next World Cup because he's pretty hungry."
Carter, who in May signed a new four-year contract with the New Zealand Rugby Union until 2015, did not rule out participating in the next tournament in England at the age of 33, saying: "Who knows?"
He suffered the injury after being named to captain the All Blacks in their final Pool A fixture against Canada while regular skipper Richie McCaw rested a foot injury.
But he has now been replaced in the squad by Manawatu back Aaron Cruden after suffering the tear during a kicking drill at Rugby League Park in Wellington.
"I knew it was pretty serious because of the pain," Carter said.
"I was actually having a bit of a shorter session than usual.
"I normally kick a good 15, 20 balls at captain's run, but I was only having four this time around.
"It was my fourth kick and I didn't know what happened.
"For something like this to happen has been really tough. I can't put my finger on why. I'm constantly asking 'why did this happen?'
"I don't have the answer, unfortunately."
The Crusaders fly-half now says he has come to terms with the blow, but experienced an emotional rollercoaster.
He added: "Lying in bed on Saturday night was pretty tough, looking back at one of the craziest days of my life.
"To be named All Black captain - something very special - and to have that taken away from you through an injury and then later to find out that my dream of being involved in the World Cup was now over.
"It's pretty gut-wrenching and disappointing what happened, but I have to get over that and continue to think positively and try to help the guys in whatever way I can."
Hansen was optimistic on the other injuries in the New Zealand squad and said McCaw "looks a million dollars".
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