The Prince of Wales and the Duke of Cambridge also congratulated the couple on social media.
The Queen's 11th great-grandchild was born on Friday at a hospital in Santa Barbara, California.
Lilibet - the Queen's family nickname - was coined when then-Princess Elizabeth was just a toddler and could not pronounce her name properly.
Her grandfather, King George V, would affectionately call her Lilibet, imitating her own attempts to say her name. It soon stuck and she became Lilibet to her family from then on.
Prince Harry and Meghan said their baby's middle name, Diana, was chosen to honour her "beloved late grandmother", the Princess of Wales.
Buckingham Palace said: "The Queen, the Prince of Wales and the Duchess of Cornwall, and the Duke and Duchess of Cambridge have been informed and are delighted with the news."
Prince Charles, Prince Harry's father, and the Duchess of Cornwall tweeted: "Congratulations to Harry, Meghan and Archie on the arrival of baby Lilibet Diana. Wishing them all well at this special time."
Prince William and the Duchess and Cambridge said: "We are all delighted by the happy news of the arrival of baby Lili."
Prime Minister Boris Johnson sent his "many congratulations" to the couple and Labour leader Sir Keir Starmer welcomed the "wonderful news".
Lilibet was born at 11:40am local time, weighing 7 lbs 11oz.
She is eighth in line to the throne - meaning Prince Andrew, who was born as second in line in 1960, moves down to ninth place.