Meanwhile presidential candidate Joe Biden embraced his wife Jill with a hug.
Viewers on social media said the Trumps looked "uncomfortable and awkward".
"Did anyone else notice the 'coldness' in the way #MelaniaTrump greeted her husband vs the warmth and embracing hug #JillBiden greeted her husband with? There's no love there and she stands up there putting on a fake smile. So cringe-worthy," wrote one person.
Others praised Mrs Trump's look, saying she looked "polished and structured" in her power suit.
Dr Biden chose an outfit she was first snapped in back in October 2017 when she attended a fundraising event with her husband in New York City.
Both wore face masks for the event but Mrs Trump took hers off to greet her husband.
The First Lady was joined by her stepdaughters Ivanka and Tiffany, who were joined by Eric Trump Jr's wife Lara, and Don Jr's longtime partner Kimberly Guilfoyle.
The group was pictured together before the event kicked off, with Ivanka also sharing a selfie with her stepmum on Twitter.
Ivanka wore a stunning cream pantsuit by Gabriela Hearst worth a whopping $4875 (U$3210), accompanied by a pricey Chanel handbag.
On Sunday, Dr Biden told CNN that her 77-year-old husband was ready to take on Trump, 74, in the opening debate.
"One of the things I'm excited for is when the American people see Joe Biden up there on stage, they're going to see what a president looks like," she said.
"Someone who is calm, steady, strong, resilient."
