The 27-year-old finished second in 4:08.74 in the second heat to keep her medal-winning dreams alive.
The Morpeth Harrier, from Lesbury, ran a calm, comfortable and controlled race, doing just what was required to book her spot in the final.
Speaking to the BBC afterwards, Weightman, who is coached by Steve Cram, said: “I always get nervous with the heats. I have run well this year, but I want to be competitive here and I wanted to get the heats out of the way.
“But it was a nice, sensible, relaxed run and I am through to the final and I can’t wait, I am really excited.”
Fellow British runner Laura Muir is also through to the final, after winning the first heat in 4:09.12.
Sunday’s final starts at 7pm (GMT) and the BBC will have coverage of the race.