"I've never felt worse in my life," he said. "My heart was like 167 beats per minute. I fainted a couple of times. I called 911 twice."
Savchenko believes he contracted the virus after flying to Texas for work, though he didn't immediately test positive when he started feeling ill.
"It was pretty bad. I came back [from the trip] and I felt like crap," he said. "I went [to get tested] and it was negative. I went the second time, it was negative. I was dying. Literally dying. I felt like I couldn't breathe. It was just bad."
"I went a third time and they were like, 'You're positive,'" Savchenko continued. "I was home for two weeks. It was bad. It was the worst."
Shortly after he recovered, Savchenko got vaccinated, something he encouraged others to do, too. "Everybody who's not vaccinated, go get vaccinated 'cause that sucks," he said. "It sucks."
While his bout with COVID-19 was unpleasant, things are going well for Savchenko on a personal front, as he's started dating model Elena Belle amid his divorce from Elena Samodanova, with whom he shares Olivia, 11, and Zlata, 4. Belle is also a mom to a 4-year-old daughter, Luna.
Last month, a source told ET that the pair met over the July 4th weekend, adding, "They bonded over their kids, as they both have children around the same age. Gleb likes that Elena is fun, but also gets the responsibilities of being a parent."
"I'm seeing someone," Savchenko confirmed on the podcast, before revealing that his kids have met her. "She's a single mom. She has a daughter [who's] four years old. We've had a couple of playdates. It's been a lot of fun."
Savchenko's life is about to get a lot busier, as he's gearing up for the new season of DWTS, which will be his first as a single dad.
"It's so scary," he said of being a working single father. "... We share custody 50/50, so a week with Elena, a week with me. I interviewed several babysitters, so literally I'm just going to try to do it all."
"I'm still going through this whole thing and it's super new to me. I'm scared as f**k, to be honest with you. I'm like, 'How's it going to work?'" Savchenko continued. "I enjoy being with my girls. To me, I'm the best dad. I give them everything. I love, love, love being with them."
Savchenko said that he plans to try and plan rehearsals with his partner around his kids' schedule as much as possible.
"I don't want to miss out on the good stuff," he said. "... Work is work. I'll do what I've gotta do."
The new season of Dancing With the Stars will premiere Sept. 20 on ABC.