The 25-year-old Morpeth Harrier, from Lesbury, near Alnwick, took command of the race in the first lap, which had begun at a very slow pace.
She led the field for the next couple of laps before four athletes passed her - Kenyan Faith Chepngetich Kipyegon, Dawit Seyaum, from Ethiopia, US runner Shannon Rowbury and Ethiopian Besu Sado.
But Laura was determined to make the final and, with the first five qualifying, she gritted her teeth and courageously hung on to fifth place, despite sustaining a nasty spike injury to her shin. She crossed the line in 4mins 5.28secs.
Fellow Brit Laura Muir also qualified for the final, comfortably finishing third in the second semi-final in a time of 4:04.16.
The final is at 2.30am GMT on Wednesday and will be shown live on the BBC.
The Games in Rio, Brazil, are Weightman’s second taste of the Olympics, after she finished 11th in the 1500m final at London 2012.