The 25-year-old Morpeth Harrier, from Lesbury, near Alnwick, crossed the line in a time of 4mins 8.37secs in Heat 2, inside Rio’s Olympic Stadium.
The top six in each heat qualified automatically, with the six fastest losers taken from all of the heats also going through. Heat 2 was won by Sifan Hassan, in 4:06.64.
Immediately after the race, she told the BBC she was hopeful the performance would be enough to put her through to the semis.
"Fingers crossed that should put me through," said Laura. "It was a messy, scrappy race and I made a few silly mistakes."
Before the race, her coach Steve Cram declared her heat the most difficult of the three in his commentary for the BBC. Three of the field clocked faster times than Laura this season.
Fellow Brit Laura Muir eased into the semi-finals, finishing third in Heat 3 in a time of 4:06.53.
Both Lauras will now compete in the semis (starting Monday, 1.30am GMT), with the aim of making the final (Wednesday, 2.30am GMT). Both races will be shown live on the BBC.
The Games in Brazil are Weightman’s second taste of the Olympics, after she finished 11th in the 1500m final at London 2012.