Weightman overtook three competitors in the final 100m as she surged to a fourth-place finish in her semi-final in 4:05.63 to claim automatic qualification.
It is the first time the former Leeds Beckett University student has reached the final at a World Championships, although she has twice progressed into the showpiece at the Olympic Games.
"I am absolutely thrilled," Weightman said.
“That was phenomenal. I knew that it was going to be really tough and that I would have to run the race of my life. I think I have just done that.
“I don't think words can summarise what it is like to compete out there in that stadium.
The 25-year-old, coached by former men's world 1,500m record holder Steve Cram, will be joined in Monday's final by British record holder Laura Muir who finished runner-up in the second heat.
Muir is arguably Great Britain's brightest podium prospect this week and tracked Olympic champion Faith Kipyegon to cross the line in second place in four minutes 3.64 seconds.
Behind her were South Africa's Caster Semenya, dominant over 800m but still something of an unknown quantity over the longer distance, and world record holder Genzebe Dibaba, evidence of how loaded the race was.
Both qualified, Semenya comfortably in third, but Dibaba having to rely on a fastest loser spot after trailing home sixth in 4mins 5.33secs.
Muir, seventh at last year's Olympics in Rio, will return for the final at the London Stadium on Monday night and said: "I'm really happy. That was a really tough semi-final, so I'm just glad to have got through it.
"That was really strong, it could easily have been a final.
"I was boxed in a bit, but I knew not to panic. I was running on the inside, not wide. I think I've conserved as much as I can.
"I kept glancing up at the screen to make sure I didn't have too big a gap. I didn't want to waste too much energy.
"But the likes of Semenya and Dibaba were behind me and they are very strong sprinters, so I thought I wanted to keep out of harm's way and have a bit up my sleeve in case people came rolling past."
In the end Muir crossed the line 0.16secs ahead of South African Semenya.