The couple were married in Murray's home town of Dunblane last April and announced the pregnancy in the summer.
Confirming the news, the tennis star's agent said: "Andy and Kim had a daughter in the early hours of Sunday morning and the family are doing well."
Shirley Erskine, Andy Murray's grandmother said she was delighted at her great granddaughter's arrival.
Andy's mother Judy tweeted a photograph of Andy Murray's gold post box in Dunblane which has been adorned with pink ribbons, with the words "Dunblane baby love.......xxx".
The post box was painted gold when Andy won the gold medal at the London Olympics.
She later tweeted another picture of tennis balls and a dummy, with the caption "Tennis family #cute".
In the run-up to last month's Australian Open, Murray, said his family would be a "priority".
The world number two made it clear that he would have cut short his first grand slam of the year and flown home if the baby had arrived earlier than expected.
He said: "My child is more important to me, and my wife is more important to me, than a tennis match.
"It's a big change for me and my wife, but that's the current priority and I'll see after that.
"I have no idea how that will change things. I still love tennis."
Speaking from her home in Dunblane, Mrs Erskine said: "I am absolutely delighted, as you can imagine. It seems a long time that we've been waiting for this but all's well now as far as I know.
"I don't have any details I just know that I've got a great granddaugher so that is just lovely.
"All I could think of after Andy had finished playing his final [in the Australian Open] was 'Oh good, at least he'll get home now'.
"It's not easy getting home from Australia. He couldn't have been further away. Fortunately, he was home in plenty of time and all's well."
Mrs Erskine said the first she heard was a text saying "she's doing well" as she had missed the first multiple recipient text Andy had sent.
"I thought 'Oh something's happened'," she said. "So, I frantically texted him back, saying 'What's happened?, what's happened?'.
She said Andy then apologised as he realised she had not received the first message as she had an old phone.
"We didn't know what the baby's sex was going to be. Andy and Kim said from the outset that they didn't want to know."
Andy's other grandmother Ellen Murray, 82, said the tennis star phoned her on Monday to give her and husband Gordon the news.
Speaking from her home in Kilsyth, she said: "Andy phoned yesterday and said 'congratulations, you're a great granny'.
"He told me the baby was 8lb 10oz when she was born."
"It's wonderful news," she added. "We watched him in the Australian Open final and I think he just wanted to come home.
"I don't really know about her name yet but I hope we'll see them all soon. I'll be looking forward to a cuddle.
"Gordon is very happy too. They did a good job."
After being beaten in the final of the Australian Open, Murray tearfully told his wife he would be on "the next flight home".
Speaking to the crowd after the defeat, he gave a message to Kim, who was "watching back home".
He said: "You have been a legend the last two weeks, thank you so much for all of your support. I'll be on the next flight home."
Following news of the birth, sports personalities and politicians tweeted their congratulations, along with hundreds of members of the public.
Sports broadcaster Gabby Logan said: "Kim and Andy Murray have had a baby girl -huge congrats."
And in a note to Murray's mother Judy, the presenter added: "@judmoo - great news for Fed Cup team 2036..."
Presenter Annabel Croft, a former British number one tennis star, wrote: "Congratulations Kim & @andy-murray on the birth of their baby daughter. Fantastic news."
Scottish First Minister Nicola Sturgeon tweeted: "Lovely news. Congratulations to @andy_murray and Kim on the birth of their daughter. Wishing every happiness."
Scottish Labour leader Kezia Dugdale tweeted: "It's a girl! Huge congratulations to Kim Sears and @andy_murray on their wonderful news."
The couple, both 28, met as teenagers in 2005 through tennis circles. They live in Surrey and married in Dunblane last April, cheered on by a large crowd of well-wishers.
Kim's pregnancy was confirmed publicly in August after they had told family and friends the news, following the 12-week scan.
Murray, the British number one and a double grand slam winner, has been crowned BBC Sports Personality Of The Year twice.
Tennis star Andy Murray's wife Kim Sears has given birth to an 8lb 10oz  girl, who arrived on Sunday morning.