Spence, the reigning individual world champion, who finished 21st at the Olympics in London last summer, was at her best as she finished top of the standings for qualification group B.
She started in eighth after clocking 2:12.48minutes from the 200m freestyle in the pool before winning 19 of her 33 fencing bouts to go into the combined run-shoot in sixth.
Spence produced a strong run-shoot time of 13:37.47 to cross the finish line in first while Burke did enough to join her in the final after placing 19th overall in the same qualification group.
Burke was fifth after posting 2:10.53 in the pool and then second after 21 victories in the fencing but came home 57 seconds behind Spence in 19th in the combined run-shoot.
That meant she failed to get into the 12 automatic qualification places but it was good enough to earn her one of the 12 highest points scorer spots as Murray missed out altogether.
Murray, who claimed Britain's 65th and final medal at London 2012 last summer, produced the fastest swim of the day, 2:08.55, to improve to eighth after placing 19th following the fencing.
However she used up the full 50 seconds on two of her four visits to the shooting range during the combined run-shoot and ended her campaign in 23rd to miss out on the final.
- Sports & Recreation