Kostas Manolas, Daniele De Rossi and Lorenzo Pellegrini scored as Roma secured a fifth win in six league outings on a night when those at the Stadio Olimpico paid tribute to Astori, a former Roma player.
Fans in the Italian capital held up banners reading "Ciao Davide" and "It was great to have you with us", while players from both sides linked arms during an emotional tribute before kick-off.
Roma goalkeeper Alisson made excellent first-half saves from Torino's Iago Falque and Afriyie Acquah, and the hosts made the breakthrough on 56 minutes when Greece international Manolas powered in a header from Alessandro Florenzi's cross.
Manolas gestured towards the sky in honour of Astori, who spent the 2014-15 season at Roma, before De Rossi doubled the lead on 73 minutes with his first club goal since May.
The 34-year-old Italy veteran was one of thousands of mourners who gave Astori, a defender who played 14 times for his country, an emotional send-off at his funeral in Florence on Thursday.
Pellegrini added a third in stoppage time with a thumping drive as third-placed Roma pulled four points clear of city rivals Lazio, turning their focus now to next week's Champions League last-16 second leg against Shakhtar Donetsk.