The 'Five Colours In Her Hair' singer shared a very sweet snap via Instagram showing little Buzz's fingers clasping on to his thumb.
"Buzz and I," read the caption.
Tom and wife Giovanna became first-time parents on 13 March, with the happy couple confirming the news - and Buzz's unusual name! - via Twitter the following morning.
"Yep, @MrsGiFletcher gave birth to our insanely gorgeous son yesterday at 7pm," wrote Tom. "He is healthy, beautiful and I've already changed his nappy."
Giovanna added: "THRILLED to announce the safe arrival of our little BOY! Born last night at 7pm. I can't stop staring at and kissing him! I'm in love!"
Since then, it's been clear Tom is relishing his new role as a dad, tweeting: "I've been having a cuddle with Buzz for about an hour. Best cuddle of my life. Ever! … I'm so so so so completely in love with my son."
He also joked: "Buzz is now one day old. He's so grown up already. Where does the time go?!"
Meanwhile, an equally thrilled Giovanna tweeted: "Thank you to everyone who has messaged over the last few days. It's been totally overwhelming. Love our little man so much."
Tom and Giovanna announced their pregnancy in October 2013. They married in London the previous year.
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Tom and Giovanna aren't alone! More stars who've welcomed babies in 2014 below:
Copyright: Rex Features WENN/Matthew Taylor/Daniel Deme/Josiah True/Matthre Taylor/Joe