The Jets appeared destined for victory when Adam Taggart's 14th goal of the season put the visitors ahead after 61 minutes.
But a stunning free-kick from Nebojsa Marinkovic in the 84th minute sparked new life into Glory and the home side quickly followed it up only a minute later with another goal from Chris Harold.
It snapped a 10-game winless run for Glory and continued the Perth club's dominance over Newcastle in recent times.
The Jets have not beaten Glory in their past nine attempts and their only victory in Perth came during the inaugural season of the A-League.
Seventh-placed Newcastle trail Sydney FC by one point in the table with two rounds remaining and are in danger of missing the finals for a fourth straight season with tough matches against Melbourne Victory and Adelaide United to come.
Glory remain bottom of the table, but sits level with Melbourne Heart on 25 points.
Sunday's other action saw Adelaide United take a huge step towards the finals with a narrow 1-0 win at Wellington Phoenix, sealed by Marcelo Carrusca's second-half penalty.
