Sinner dominated the first set 6-2 in the last 16 tie but then had to remain patient as Cobolli dug deep to clinch the second set 7-6(4).
Four-time Grand Slam champion Sinner is playing in Austria after retiring with cramps earlier this month in his third-round match against Tallon Griekspoor in the Shanghai Masters.
Sinner did not face a break point in the first set against Cobolli, but was unable to convert any of the four break opportunities he secured in the second.
"He is a great talent and a great competitor, and we both played some great tennis at times," said world number two Sinner of his compatriot.
"I had some chances in the second but could not use them, and that is tennis.
"I tried to stay in the moment and play with a great mentality, and I am very happy with today's match," he added.
Sinner will face eighth seed Alexander Bublik in the quarters after the Kazakh downed Francisco Cerundolo 6-4, 6-2 in the last 16.
Second seed Alexander Zverev is also through to the last eight, where he meets Griekspoor.
France's Corentin Moutet gained swift revenge on Daniil Medvedev.
The pair met in last Sunday's final in Almaty, with Medvedev prevailing.
Four days later, Moutet emerged the winner over the Russian 7-6(3), 6-4.