Hewitt is a four-time Queen's champion and this year he is bidding to become the oldest man to win the pre-Wimbledon warm-up event, surpassing American legend Jimmy Connors, who lifted the trophy aged 30 years and 284 days in 1983.
He will fancy his chances after seeing off world number eight del Potro to claim his first win over a top 10 player since he defeated Juan Monaco in Valencia last October.
The former world number one, who made his first appearance at Queen's as a 17-year-old in 1998, is well into the twilight of his career and his lowly spot at 82 in the world rankings reflects his diminished status.
But the 32-year-old former Wimbledon champion enjoyed notable victories over highly-regarded Grigor Dimitrov and former Queen's champion Sam Querrey en route to the last eight.
It was the first time Hewitt had won three successive matches at a tournament since his run to the final in Newport, also on grass, last July.
Aided by two del Potro double faults, Hewitt earned two break points at 3-2 and took the first set in style, unloading a forehand winner that flashed past the Argentine.
He finished off the set in style, breaking again as he nailed a volley on set-point.
Del Potro showed his grass court pedigree when he won the bronze medal in the London Olympics at Wimbledon last year and he hit back emphatically with two breaks in the first three games of the second set.
That was enough to take the set, but Hewitt broke for a 3-1 lead in the decider.
Hewitt's deep groundstrokes were keeping the towering del Potro off balance and he broke again to seal the win before inviting his son Cruz onto the court to celebrate the victory with him.
Hewitt's semi-final opponent is Marin Cilic, who remains on course to retain the Queen's title after the defending champion defeated Czech second seed Tomas Berdych 7-5, 7-6 (7/4).
Cilic won in bizarre circumstances last year when his final opponent David Nalbandian was disqualified for kicking an advertising board into the shins of a line-judge in frustration at losing a point.
The Croatian fifth seed is now just two wins away from becoming the first player to retain the trophy since Andy Roddick in 2005 after a dominant display against world number six Berdych.
"I served really good in the crucial moments and got free points. I'm really happy to be through," Cilic said.
Cilic had lost four of his previous five meetings with Berdych and had never managed a grass court victory against a player ranked in the top 10.
But the 24-year-old edged a tight first set when he landed the only break of serve at 6-5.
Cilic, who brought an end to his nine-year partnership with Australian coach Bob Brett just before the recent French Open, held his nerve when the second set went to a tie-break and sealed one of his best wins in 2013.
Results from the fifth day of the ATP tournament at Queen's Club (x denotes seeding):
Marin Cilic (CRO x5) bt Tomas Berdych (CZE x2) 7-5, 7-6 (7/4)
Lleyton Hewitt (AUS) bt Juan Martin del Potro (ARG x3) 6-2, 2-6, 6-2
Andy Murray (GBR x1) bt Benjamin Becker (GER) 6-4, 7-6 (7/3)