The Wishaw 43-year-old drew on his fighting qualities to repel the Australian’s fight back from a 9-7 deficit.
Twice Robertson got within a frame but Higgins kept his nose in front and finished in style with a 101 clearance in the last. He’ll face David Gilbert for a place in the final.
Earlier, Judd Trump took a swipe at Stephen Maguire after demolishing the Scot 13-6 to reach the semi-finals.
The 29-year-old mocked the Scot’s pre-match claim that he was a “lucky” player by joking: “I got a lot of luck out there – I think that was the only reason I did win.”
Trump added: “People take it differently. Some people applaud their opponent and some people brush it off like it wasn’t their fault – so maybe that’s just his way of looking at it.”
Trump, who is targeting a place in his first world final since 2011, had built on his stunning opening session by winning the first two frames yesterday and establishing a 9-1 lead. Maguire managed to claw his way back to respectability by taking four of the next five frames including a break of 110 but Trump swiftly responded with breaks of 68 and 82 to get over the line .
Shrugging off the fact that he is the new odds-on favourite for the title, Trump added: “It’s always good to beat anybody that comfortably but there is a long battle ahead.
“It was nice to be playing with no real pressure out there and just being able to fully enjoy it with that kind of lead.”
Asked to sum up his performance in a word, Maguire replied: “S**t” – which will earn him a £250 fine from the governing body for swearing in a press conference.
Trump will face qualifier Gary Wilson whose run continued as he converted a 9-7 overnight advantage into a 13-9 win over two-time finalist Ali Carter. The Wallsend 33-year-old endured six years off the tour and took jobs as a taxi driver, a barman and a line worker in a frozen food factory before grasping his second chance.
Wilson said: “I was desperate to get back on the tour and just scrape a living. So to go from that point to where I am now, in the one-table set-up, is absolutely amazing and the sort of thing you dream of.”
Gilbert held off last year’s semi-finalist Kyren Wilson to claim a 13-8 success.