The fifth-ranked Medvedev triumphed 6-7 (7/9), 6-4, 7-6 (7/4), 2-6, 6-3 to avenge his Australian Open final loss to Sinner in January.
His four-hour victory also ended a five-match losing streak against the Italian as Medvedev reached his ninth Grand Slam semi-final.
The 28-year-old former US Open winner will face either defending champion Carlos Alcaraz or Tommy Paul of the United States for a place in Sunday's final.
"I knew if I wanted to beat Jannik it needs to be a tough match. He's not a guy you can beat easy anymore," said Medvedev.
"I felt at one moment he was not feeling that good but he started feeling better so I'm happy I managed to still stay at a high level."
Medvedev added: "It's always tricky because you want to play more points to make him suffer a little bit more but in a good way."
In a tight first set in which neither man gave up a break point, Medvedev edged ahead in the tie-break on the back of a lung-busting 33-shot rally.
However, he squandered a set point and his game suddenly fell apart, serving up a double fault as Sinner pocketed the opener.
The mercurial Russian carved out the only break of the second set in the third game on his way to levelling the quarter-final.
Sinner slipped another break down at 2-1 in the third set and immediately summoned the tournament physio.
The Italian, appearing distressed, had his blood pressure taken courtside before he left for a medical time-out.
After a 10-minute delay, the 22-year-old resumed play and thrilled the Centre Court crowd when he bravely clawed his way back to 5-5.
He then wasted two set points which allowed Medvedev to sweep through the tie-break.
However, the Italian was undaunted and levelled the quarter-final on the back of breaks in the third and fifth games of the fourth set.
But it was Medvedev finished the fresher of the two players, breaking in the final set on his way to victory.
