The big Argentine was looking to set up a mouth-watering quarter-final against Rafael Nadal, looking to avenge his defeat by the Spaniard, seeded second at Wimbledon, in the French Open semi-finals last month.
Del Potro saved two break points in the sixth game, a deft chip from the baseline that dipped over the net followed by a forehand winner, before taking the opening set in a 7-1 tiebreak.
Simon, looking for his third Grand Slam quarter-final at the age of 33, secured the first break of the match to take a 4-3 lead in the second set when Del Potro netted a return.
But the Argentine broke back immediately with a volley and then saved another break point at 5-5 with an ace.
The 12th game went to deuce eight times, with Simon saving four set points, as Del Potro failed to take his chances, the worst when he put an easy volley over the baseline with the court wide open.
Del Potro, who made more than 20 unforced errors in the set, made hard work of winning the tiebreak, clinching it 7-5 on the seventh set point when Simon put a return high and wide and the Argentine roared in celebration.
Del Potro fell 4-1 behind in the third set after Simon broke in the third game.
The Argentine, who leads the stubborn baseliner 4-3 in previous encounters, levelled on his second break point and it went to 5-5, but Simon broke again to take the first set off Del Potro in the tournament so far.
(Reporting by Rex Gowar; Editing by Kevin Liffey)
