Dzeko's remarkable goal in the 89th minute rewarded Roma for their persistence after they hit the woodwork three times although the battling hosts also struck the post and had a goal disallowed.
The game appeared to be heading for a draw until substitute Kluivert, the son of former Barcelona and Netherlands forward Patrick Kluivert, collected the ball in midfield, slipped past two Torino defenders and forced his way to the byline.
The 19-year-old crossed deep into the area where Dzeko met the ball with the inside of his left foot and sent a deft volley into the far corner of the net from a seemingly impossible angle.
Kluivert was signed from Ajax during the close season.
Aleksandar Kolarov and Dzeko both hit the post during the first half and Dzeko again struck the woodwork from a free-kick in the second half.
Torino midfielder Tomas Rincon also struck the woodwork in the first half with a dipping shot and Iago Falque had a goal disallowed for offside after a VAR review three minutes after the break.
Midfielder Javier Pastore and goalkeeper Robin Olsen also made their debuts for Roma, who were UEFA Champions League semi-finalists last year.
Torino coach Walter Mazzarri was sent off for protesting about what he thought should have been a penalty for his side.
