Higgins led 9-7 coming into the final session after a four-frame salvo earlier in the day had completely changed the course of the tie.
He opened up with a break of 81 to make it five in a row before Robertson finally got on the board with breaks of 50 and 114.
The pair exchanged the following three frames as Higgins edged closer to victory and both players tried to capitalise on mistakes made by the other.
However Higgins had the greater fluency and when he got in during frame 23 it felt like game over and it proved to be the case.
'This is what you call a classy finish!' - Higgins seals victory over Robertson in style
The Wizard of Wishaw finished off superbly with a wonderful break of 101 to book his place in the last four where he’ll face David Gilbert.
Gary Wilson became the fifth lowest-ranked player in history to reach the World Championship semi-finals with a 13-9 win over Ali Carter at the Crucible Theatre.
The qualifier, the world number 32 from Tyneside, returned with a 9-7 advantage, building on it immediately with a break of 117 in the opening frame.
Two-time finalist Carter sliced his lead to a solitary frame by taking the next two, helped by a knock of 128, but Wilson flipped the momentum and won three frames on the spin to progress.
Wilson will face Judd Trump in the last four after the Masters champion rounded off a 13-6 win over Stephen Maguire. Trump held an 11-5 lead after their second session on Wednesday afternoon, and finished off breaks of 68 and 82 after Maguire rolled in a 122 to close to 11-6 behind.
Leading 7-1 overnight it looked as if Trump might wrap things up quickly when he won the first two frames of the session to move 9-1 clear.
However, Maguire dug deep and won four of the following five frames, including a break of 110 to force a final session.
Gilbert is through to the semi-finals for the first time in his career as he saw off Kyren Wilson 13-8.
Gilbert led 10-6 from the previous night's session but Wilson battled back a bit by winning two of the first three frames to move to within three of his opponent, but the 16th seed steadied himself as knocks of 56, 60 and 63 carried him into the last four.
Gilbert is the man who recorded the 147th professional 147 and is now guaranteed the biggest payday of his career of at least £100,000.
Quarter-finals (best of 25 frames)
- David Gilbert (Eng) 13-8 Kyren Wilson (Eng)
- John Higgins (Sco) 10-8 Neil Robertson (Aus)
- Gary Wilson (Eng) 13-9 Ali Carter (Eng)
- Judd Trump (Eng) 13-6 Stephen Maguire (Sco)
Second round results (best of 25 frames)
- Mark Williams (Wal) 9-13 David GIlbert (Eng)
- Barry Hawkins (Eng) 11-13 Kyren Wilson (Eng)
- John Higgins (Sco) 13-11 Stuart Bingham (Eng)
- Shaun Murphy (Eng) 6-13 Neil Robertson (Aus)
- Mark Selby (Eng) 10-13 Gary Wilson (Eng)
- Ali Carter (Eng) 13-9 Zhou Yuelong (Chn)
- Judd Trump (Eng) 13-9 Ding Junhui (Chn)
- Stephen Maguire (Sco) 13-12 James Cahill (Eng)
First round results (best of 19 frames)
- Mark Williams (Wal) 10-7 Martin Gould (Eng)
- David Gilbert (Eng) 10-7 Joe Perry (Eng)
- Barry Hawkins (Eng) 10-1 Li Hang (Chn)
- Kyren Wilson (Eng) 10-4 Scott Donaldson (Sco)
- John Higgins (Sco) 10-7 Mark Davis (Eng)
- Stuart Bingham (Eng) 10-9 Graeme Dott (Sco)
- Shaun Murphy (Eng) 10-0 Luo Honghao (Chn)
- Neil Robertson (Aus) 10-1 Michael Georgiou (Cyp)
- Mark Selby (Eng) 10-7 Zhao Xintong (Chn)
- Luca Brecel (Bel) 9-10 Gary Wilson (Eng)
- Jack Lisowski (Eng) 6-10 Ali Carter (Eng)
- Mark Allen (NI) 7-10 Zhou Yeulong (Chn)
- Judd Trump (Eng) 10-9 Thepchaiya Un-Nooh (Tha)
- Ding Junhui (Chn) 10-7 Anthony McGill (Sco)
- Stephen Maguire (Sco) 10-9 Tian Pengfei (Chn)
- Ronnie O'Sullivan (Eng) 8-10 James Cahill (Eng)