Barlow, whose fourth child Poppy was tragically delivered stillborn last month, said that he was thrilled for his bandmate, reported Sun online.
"Massive congrats Rob and Ayda. So happy for you," Barlowe posted on Twitter.
Williams' best friend Jonathan Wilkes also took to Twitter to express his delight at the new arrival.
"Just had the best news," he wrote.
Field went into labour eight days early, with the baby - who will be known as Teddy - born at London's Portland Hospital at 3.33pm and Williams was present for the delivery. She weighed 7lb 4oz.
The 38-year-old singer and 33-year-old Field began dating in 2006 and married in August 2010.