The Spanish top seed seemed to be cruising after racing a set and 4-1 up, but Nardi reeled off five games in a row to send the match to a third set.
Alcaraz grabbed the decisive break in the fourth game of the final set and wrapped up a 6-1, 4-6, 6-3 victory.
"He played a few good points and then it seemed like he started to play like the world number one," Alcaraz said of Nardi's mid-match recovery.
"I couldn't do anything, I just tried to stay there, stay strong mentally. My energy levels probably went down a little bit, but credit to him. I'm really happy (that I) forgot everything and restarted in the third set."
The four-time Grand Slam champion, who is making his debut in Doha, meets the Czech Republic's Jiri Lehecka for a place in the last four on Thursday. Lehecka swept aside Hungary's Fabian Marozsan 6-4, 6-2 in barely an hour.
Second seed Alex de Minaur booked his place in the quarter-finals of the Qatar Open with a controlled 6-4, 6-4 victory against Botic van de Zandschulp on Wednesday.
The Australian, who chalked up his 200th tour-level hard-court win in his first-round victory against Roman Safiullin, showed off his pace around the court as he brushed past the Dutch qualifier in two hours, saving all four break points he faced.
It is the third time in successive tournaments that De Minaur has reached the quarter-final stage.
At the Australian Open, he bumped up against eventual winner Jannik Sinner in the last eight and in Rotterdam earlier this month he went all the way to the final before succumbing in three sets to Carlos Alcaraz.
Chasing his first trophy of the season, De Minaur, ranked No.8 in the world, now faces the Russian Andrey Rublev who beat Nuno Borges 6-3, 6-4.
A Doha winner in 2020, Rublev has also been on a decent run of late, reaching the semi-finals in Montpellier and quarters in Rotterdam.
Fourth seed Daniil Medvedev, the 2023 champion, powered past Zizou Bergs 6-2, 6-1, while Britain's Jack Draper won in straight sets against Christopher O'Connell.
Felix Auger-Aliassime received a walkover as Hamad Medjedovic withdrew after hurting his leg in the previous round when he beat Stefanos Tsitsipas.