The Oscar-winning actress, 63, worked alongside Rickman throughout his career, including playing his wife in the romantic comedy Love Actually.
A collection of the Harry Potter star’s diaries have been published following his death in 2016 aged 69, giving readers an intimate look at his life and career.
In an interview on BBC Breakfast on Thursday, presenter Charlie Stayt noted one of the diary entries from 1994 said: “Leave a rude note for Emma.”
Dame Emma said the entry must have been written during the pair’s time in Los Angeles and how, on their flight home, they had laughed non-stop.
She said: “I remember him telling a story about him being very close to a cat and someone who’d farted very loudly and it had parted the cat’s hair.
“I just remember him telling that story in business class on British Airways and sort of doing that thing of going backwards and forwards.
“He was so funny, Alan. That’s the thing. And you don’t necessarily get that. All of his performances are so extraordinary but when he’s funny, oh, my God, is he hilarious.”
The veteran actress said Rickman could also be “terrifying”, adding: “That’s fine because he had such dignity, Alan, and enormous intelligence and he was not a frivolous person in any way at all.”
Dame Emma said the pair could get “quite giggly and silly” together at times, like on New Year’s Eve, but in general he was a “very deep thinker”.
“He took the world very seriously and his role in it very profoundly seriously, which is why he was a great artist, actually”, she said.
The 2003 film Love Actually has become a Christmas staple and the scene where Dame Emma confronts Rickman’s cheating character has been lauded as a heartbreaking moment of cinema.
Reflecting on their time filming the movie, Dame Emma said: “Alan and I had energies that were almost entirely opposite.
“His was subterranean and mysterious and dark and mine was canine and above-ground and it ebullient. So it was like having Eeyore and Piglet on set, basically.
“And that’s what it was like with Love Actually. I’d be bouncing around doing Piglet and he would just about put up with my kind of small pinkness.”
During the interview, Rickman’s wife Rima Horton said fans would often quiz him about his roles, mainly Professor Snape in the Harry Potter films and villain Hans Gruber in Die Hard.
She said his diaries show she was able to “stabilise” him.
She said: “He says quite a few times in the diaries something that I didn’t know he’d written – ‘and then Rima turned up and I calmed down’.
“So I think I did have that role in his life of keeping him, I suppose, stabilised. Because Alan was a classic Pisces, swimming in two directions at once, and I think as I am an Aquarian that I was quite a good balance to that.”