Potter led early on, but unheralded 23-year-old Spain-based Andrews came through strongly to win in Highgate by beating her personal best by 83 seconds.
“I felt good and was confident to hold that position at that stage,” said the Scot, who clocked clocked 32.05.4, nearly 10 seconds inside the qualifying mark.
“But Jess had a great run; I used to train with her, so well done to her.”
Marathon trio Callum Hawkins, Tsegai Twwelde and Derek Hawkins have already been selected by GB, so Potter takes the Scottish track and field contingent to four.
Rodger Harkins, director of coaching with scottishathletics, said: “It is a fantastic achievement by Beth Potter to run inside the standard for the second time.”
“We’re thrilled at some of the recent performances by Scottish athletes – including the three guys who have made the marathon team.”
Potter’s fellow Scot Andrew Butchart moved closer to Olympic selection and broke a long-standing Scottish record in the 5,000m in Hengelo, Netherlands.
Butchart – who finished the race with one shoe – was ninth with a time of 13:13.30. That beat Nat Muir’s mark of 13:17.9 in 1980 and Butchart can clinch a place in Rio if he finishes in the top two of his event at the British Championships next month.
Compatriot Eilish McColgan can also make it to Brazil with a first or second-placed finish at the Olympic trials in the women’s 5000m.
She was sixth in Hengelo, with a time of 15:16.51.
