The two players remained on serve until the match-deciding tiebreak, which again saw del Potro jump out to an early advantage.
Del Potro, who claimed the Indian Wells title earlier this month, fed off the energy of a raucous crowd that was firmly behind the Argentine, wearing down the 20th-seeded Canadian in a match lasting nearly three hours.
"We both played well, the difference was perhaps that I just played better in the tie-breaks".
Fifth-seeded del Potro advances to the semifinals where he will square off against American John Isner.
The Canadian was defeated despite not facing a single break point in the first two sets, before the contest took a freakish twist at the start of the decider when both players were broken twice in a row.
"Del Potro is the hottest player on tour right now, hands down", Isner added.
Pedro Pascal joins 'Wonder Woman 2' in 'pivotal' role
This isn't the first time Pascal and Jenkins have worked together - they previously teamed up on the TV movie Exposed . Pedro is known for portraying the role of Oberyn Martell on " Game of Thrones " and Javier Pena on " Narcos ".
Their fears worsened as sixth-ranked del Potro called for the trainer, although after a lengthy discussion with officials no treatment was given and the second set began.
All three of Isner's wins over the 2014 U.S. Open champion have now come in ATP Masters events and once a first set tie-break was negotiated in windswept conditions, the 32-year-old confidently held his nerve to send Cilic home.
However, Raonic started to pull the strings during the rally, forcing Del Potro to stretch across the court on his return. Del Potro is coming off a victory at Indian Wells where he defeated Raonic 6-2, 6-3 in the semis.
Del Potro carved out a 5-2 lead, a position of strength from which he was finally able to see the match out.
The world number six's excellent run of form continued as he fought back from 1-4 down in the first set to beat the 22nd seed 6-4 6-2. The way the ball bounces, the court isn't too fast, so it's great for me.
"I played extremely well and every match I have gotten better and stronger", Isner said.