Sinner's sublime drop-shot winner broke world No. 73 Ivashka for a 4-3 lead in the decider and the Italian went on to seal the victory when the Belarusian's forehand sailed long on match point.
Sinner, seeded 11th, said the crowd at Louis Armstrong Stadium had helped get him across the finish line.
"The atmosphere is always very great, especially when I'm down," he said in an on-court interview.
"Thanks to you guys for keeping my head up because today I was struggling. I was not playing my best and in the fifth set, I tried to dig deep.
"I'm very happy to be in the next round."
The Italian has now reached the quarter-finals of all four Grand Slam tournaments, becoming the youngest man to do so since Novak Djokovic in 2007-08.
Ivashka exits with his head high after impressive wins over Sam Querrey (34), eight seed Hubert Hurkacz (25) and Sinner's countryman Lorenzo Musetti (20) for his deepest run in New York.
Sinner will next meet either Spaniard Carlos Alcaraz (19) or Croatian Marin Cilic (33), who are playing their fourth-round match at Arthur Ashe Stadium.