The baby was born Friday, a Clinton spokesman said.
"Marc and I are full of love, awe and gratitude as we celebrate the birth of our daughter, Charlotte Clinton Mezvinsky," Clinton's message read.
Former President Bill Clinton and former Secretary of State Hillary Clinton said they feel "blessed, grateful and so happy to be the grandparents of a beautiful girl."
"Chelsea is well and glowing. Marc is bursting with pride. Charlotte's life is off to a good start," the elder Clintons said in a statement. Charlotte was born in New York.
Chelsea Clinton waited for the birth to find out the gender of her baby. "There are so few mysteries in life ... in which any answer is a happy one," she recently told CNN's Fareed Zakaria.
Clinton and her husband, Marc Mezvinsky, announced in April they were expecting their first child.
Chelsea is Bill and Hillary Clinton's only child, and her baby is their first grandchild.
Chelsea Clinton married Mezvinsky, the son of two former members of Congress, in 2010.
Ever since their daughter's nuptials, Bill and Hillary Clinton have applied good-natured public pressure on her to make them grandparents.
In April 2012, Hillary Clinton told ABC it wasn't "really up to" her whether she would ever earn the grandma title, but added, "I would like to have that title. ... I think I'd be pretty good."
Chelsea Clinton has regularly said that her parents were "unapologetic" in their pleas for grandchildren. "(They) are not shy about saying that in public and private," she told ET in 2013.
Bill Clinton also said last year on CBS that if Hillary Clinton could be president or a grandmother, the frontrunner for the Democratic nomination in 2016 would say grandmother.
"If you ask her, I think she'd say grandmother, but I have found it best not to discuss that issue," said the former president. "My goal is to live to be a grandfather. The rest of it's out of my hands."
Own family ambitions
Chelsea Clinton has stoked some of the baby talk, too.
In an October 2013 interview with Glamour, she said she and her husband want "to start a family." "So we decided we were going to make 2014 the Year of the Baby," Clinton said.
She later told the reporter: "Call my mother and tell her that. She asks us about it every single day."
For the last month, the baby has somewhat overshadowed the Clintons' everyday activities. When Hillary Clinton, who recently published her latest memoir, "Hard Choices," would travel to book signings, a number of well-wishers mentioned her becoming a grandparent.
"I must have shaken 70,000 hands and over half of them mentioned something about being a grandparent," Clinton said last week during a panel discussion with CNN's Sanjay Gupta at the Clinton Global Initiative annual meeting.
Hillary Clinton later added: "Most of us, when we have our children, we're still younger, we're still striving, we're still preoccupied with what's going to happen in our lives, and I think a lot of people look back and say, 'I did the best I could,' but...being a grandparent you just have that freedom, at least that's what I'm told, and I'm anxious to find out."
Current Secretary of State John Kerry welcomed the news: "Being a grandparent is the best -- period," he said on Twitter.
CNN's Carma Hassan contributed to this report.