Robertson is now one of nine top-16 seeds to crash out in York, as Joyce won a thrilling final frame to win the match 6-5 and progress into the fourth round.
Robertson was clearly disappointed to crash out but also paid tribute to his opponent for a deserved win.
"I have to congratulate him. He played a fantastic match," Robertson said. "I fired in a lot of breaks and I am not disappointed with the way I played at all. He kind of had nothing to lose."
In a further blow to Robertson, he will now miss the Masters in January because he will slip out of the world's top 16.
Elsewhere, Mark Allen, Joe Perry, Lyu Haotian, Stephen Maguire and Ryan Day were also among the round-three winners on day seven at the York Barbican.