Everton loanee Kean knew little about the winner as Rodrigo Bentancur’s effort inadvertently came off his head and went in after 16 minutes.
Close to half-time, Jordan Veretout saw his penalty saved by Wojciech Szczesny after Tammy Abraham was brought down.
Abraham had put the ball in the net as he was fouled, but referee Daniele Orsata had already blown for a penalty.
The England striker grabbed the ball to take the spot-kick but had it snatched off him by Veretout, who saw his poor effort kept out by Szczesny.
Federico Bernardeschi struck a spectacular bicycle kick straight at Rui Patricio in the second half.
The Italy international could have put his side ahead after just 15 seconds but a low drive was gathered by former Wolves goalkeeper Patricio.
Kean’s fortunate goal put Juve ahead and Roma responded with Gianluigi Mancini and Abraham seeing headers saved by Szczesny, before the controversy of the penalty incident.
Roma’s third league defeat of the season means Mourinho’s side lie nine points behind Napoli, who hold a 100% record after eight games, while Juve move up to seventh after turning around their poor start.