It paid off when substitute Nego was quickest to a ball that pinged between defenders in the area, slotting home to equalise two minutes from the end of normal time and Szoboszlai struck in the second minute of added time to seal a stunning win.
Playmaker Szoboszlai was key to Hungary's effort, whipping crosses and set pieces into the box, and he almost put his side level just before the break with a long-distance free kick that required a full-stretch dive from Hannes Haldorsson to save.
The Hungarian siege on Iceland's goal continued in the second half, but they struggled to create many decent chances and looked vulnerable on the break as the visitors sought to exploit the space behind the home defence on the counter.
Iceland substitute Albert Gudmundsson almost killed the game off with a late sliding effort but just failed to connect with Jon Dadi Bodvarsson's cross and it was the costliest of misses.
Extra time loomed but Szoboszlai had other ideas and capped a superb performance by firing home in the second minute of added time to seal a stunning win and send his side through to the finals, prompting wild celebrations by his team at the end.
Euro 2020, which is being staged across the continent, has been postponed until June-July next year due to COVID-19.