The win set a new national record as the Azzurri extended their unbeaten run to 31 games, surpassing the 30-game streak posted under Vittorio Pozzo between 1935 and 1939.
Marko Arnautovic saw an effort ruled out by VAR (video assistant referee) in the second half as Austria threatened to cause an upset, but substitutes Chiesa and Pessina booked the Azzurri's last-eight spot with clinical finishes in the first half of extra-time, although Sasa Kalajdzic did set up a frantic finish with his 114th-minute goal.
Roberto Mancini's side will face the winner of Monday's (AEST) clash between Portugal and Belgium in Munich next weekend.
Italy started strongly and had seven shots before the half-hour mark, Daniel Bachmann keeping out Nicolo Barella with his feet in what was the Azzurri's best chance in that period.
At the other end, Arnautovic blazed over from a promising position, while Ciro Immobile crashed a superb effort off Bachmann's right-hand post from 25 yards.
Bachmann pawed away a low effort from Leonardo Spinazzola shortly before the interval as Italy ultimately failed to make its first-half dominance count.
Austria improved dramatically after the break and thought it had gone ahead midway through the second half, but Arnautovic's deft header was ruled out by VAR for offside.
Franco Foda's side might have thought it would have the edge in extra-time given Italy's lethargy for much of the second period, yet it was the Azzurri that struck the decisive blow.
Chiesa controlled Spinazzola's cross in the 95th minute, cut inside Konrad Laimer and lashed a fine shot past Bachmann.
The Austria goalkeeper did well to keep out Lorenzo Insigne's free-kick soon after, but there was little he could do to deny Pessina in the 105th minute, the Atalanta midfielder powering past him from six yards after skipping past Martin Hinteregger.
Kalajdzic set up a grandstand finish with a clever near-post header from Louis Schaub's corner, ending Italy's run of 1168 minutes without conceding a goal, yet Mancini’s men held firm to keep their UEFA Euro 2020 dreams alive.