Cairney, who hosts the weekend morning slot, told Digital Spy that Grimshaw's likeable personality will come across on air.
"Moyles is Moyles," she said. "No-one will ever forget what Moyles did for the breakfast show.
"He's just been such a personality and done so many amazing things, and everyone has something to say about him and I think that's great.
"And to get up at that time every morning for that long - I did it for two years on 1xtra and it really will affect your entire being, so hats off!"
She continued: "But at the same time, I was absolutely ecstatic when I heard that Grimmy had been given the job, because he's completely the right person for it.
"He's one of the funniest people I know. He has anecdotes for days, he's also just a really great broadcaster - he loves music, he has a massive sense of humour about showbusiness in general.
"I cannot wait. Everybody loves him and that's for a reason - it's because he's a really great guy. I think that that will come across on air and I think that everybody will love him as well."
On whether Grimshaw could please the BBC Trust by bringing in younger listeners, Cairney added: "I don't know what the magic is to getting a younger audience, but I definitely think that Grimmy is a funny person.
"In terms of his music tastes and his reference points and what he likes and what excites him, I think that is appealing to a youth market. And a big curly quiff!"
Asked why Radio 1 remains so popular in an age of digital radio and streaming music, she said: "You can't ever take away what radio gives you. It gives you a friend. It gives you someone laughing in the corner of your room.
"It connects the dots between all of your favourite music. It helps you wake up in the morning and it helps you relax in your house and I think only radio can do that."
Cairney added: "Radio 1 is the only place that's Radio 1. There's not really anything like it. I can't think of any station that you can compare it to. There's just so many different threads of personality there and so many different things to discover.
"A lot of people focus on the daytime shows and think one thing but as soon as you turn it on in the middle of the night you think a completely different thing. The iPlayer has only made that better.
"Some of the specialist shows are absolutely astounding. I think that all of those personalities and all that music put together and that real creativity as well - that enthusiasm bounds out of your speakers and you can't get away from it."
Gemma Cairney presented Radio 1 Stories: Bruising Silence earlier this week on BBC Radio 1. The programme is available now via BBC iPlayer.