Charvet proposed during a "summertime party" he threw on Saturday night, Burke tells PEOPLE.
"It was a total surprise," she says. "It was all very romantic." She added, "He's my best friend."
Charvet gave his new fiancée an eternity band engagement ring because "it's forever," explains Burke, adding that the two of them (who are both 34) have known each other for years – but started dating last year, during Rock Star: INXS.
At the time, Burke had just split from husband Garth Fisher, a plastic surgeon featured on ABC's Extreme Makeover. She and Garth have two daughters – Neriah, 6, and Sierra, 4 – who have on occasion come with mom to tapings of the series at Stage 36 of CBS Television City.
As for the baby to be, Burke confirms to PEOPLE that she is indeed expecting. Supernova drummer Tommy Lee had mentioned that Burke told the band the good news last weekend, when he went on a Star 98.7 radio show in Los Angeles.
From 1999 to 2002, Burke hosted E!'s Wild On! She has also made guest appearances on TV shows including Monk and Las Vegas.
Charvet played Matt Brody on Baywatch from 1992 to 1995 and Craig Field on Melrose Place from 1996 to 1998.