Former world champions John Higgins and Ronnie O'Sullivan both suffered quarter-final defeats at snooker's Grand Prix in Aberdeen on Friday.
O'Sullivan had been in superb form prior to meeting Robertson
Robertson won the first two frames against O'Sullivan and although his opponent made it 2-1, Robertson won the next three to claim victory.
Higgins, the defending champion, lost 5-2 to Mark King, while Australian Neil Robertson beat O'Sullivan 5-1.
The Scot never recovered from losing the first four frames to King.
"It went wrong all night. My concentration was all over the place from the first frame," he admitted. "That was the telling point in the match.
I kept my composure because I knew I was playing well and I knew I would be creating chances for myself
"My safety was atrocious and I was leaving chances on a plate for him. You've got to perform, but I just didn't turn up.
"I was confident about my game and even at 4-0 still felt brilliant and that I could get back, but all night I struggled."
Stoke qualifier Jamie Cope sealed a semi-final spot with a 5-4 win over Joe Perry, clinching the decider 48-39.
"I just had to hold myself together," said Cope, who had earlier knocked in breaks of 50, 62 and 53.
"If I can carry on playing alright, I can win it."
Friday's final match saw Scot Alan McManus see off Englishman Ian McCulloch 5-3.
Robertson had lost all five of his previous ranking event quarter-finals but O'Sullivan admitted: "He was on fire. He just blew me away."
The world number three added: "You have to give him credit. He didn't miss a ball.
"Sometimes you come up against someone like that - he deserved his win. I wish I could have given him a better game, but I didn't.
"You can't just turn up with your B game and think you're going win, you can't.
"Today he played fantastic. If he carries on playing like that he'll fancy his chances of winning the tournament."
Robertson added: "I will definitely say this is the best performance of my career so far.
"I'm very happy with the way I played. I've scored heavier in matches and maybe played better, more explosive stuff. But playing like that against someone like Ronnie O'Sullivan is so much harder.
"I kept my composure because I knew I was playing well and I knew I would be creating chances for myself.
"It was a great result for me. I said before the match I was probably one of the few players who can beat him when he's playing well. I definitely believed that.
"When I get in and I'm concentrating and my positional play is good, then I score almost as heavily as anyone. There was absolutely no reason why I shouldn't think I could beat him."