“We may have four crew members on board from four different nations… but we’re a united team with a common mission,” Ms Moghbeli said after the separation.
For 29 hours, the capsule will race to catch up to the ISS with a planned docking at around 8.50am on Sunday.
The launch was pushed back to Saturday to give engineers an extra day to review a component of the Crew Dragon capsule’s environmental control and life support system, Nasa said in a blog post.
It is the first space mission for both Ms Moghbeli and Mr Borisov.
“This is something I’ve wanted to do for as long as I can remember,” said Ms Moghbeli, a Naval test pilot, during a media call last month.
“One of the things I’m most excited about is looking back at our beautiful planet,” added the 40-year-old American.
“Everyone who I’ve talked to who has flown already has said that was a life-changing perspective – and also floating around in space, it seems really fun.”
Crew-7 is set to be the seventh routine mission to the orbital platform for Elon Musk’s SpaceX, with the first coming in 2020.
Nasa pays SpaceX for the taxi service as part of a commercial crew program that it put in place to reduce dependency on Russian rockets for astronaut transport after the space shuttle program ended in 2011.
Boeing is the other contracted private partner, but its program remains mired in delays and technical difficulties. It has not yet flown any crew.
Mr Borisov will be the third Russian to fly on a SpaceX Crew Dragon capsule, fixed atop a Falcon 9 rocket.