Borriello's goal on 80 minutes saved Andrea Pirlo's blushes, after he missed his second penalty in a week with eight minutes gone.
Borriello, who has been disappointing since joining on loan from Roma, came off the bench and five minutes later smashed home a Mirko Vucinic knock-down on the half-volley.
It was a seventh win in a row for Juve, who are unbeaten after 34 Serie A games this season, and they moved closer to their first Scudetto since being relegated for their role in the Calciopoli match-fixing scandal.
"He deserved that goal, because he came here really motivated but not in great physical condition," said Juve boss Antonio Conte.
"He had to work to get himself back into shape... and now he's up with everyone else."
Conte added: "This season we have created a base for the future: we're returned to being competitive, we've done extraordinary work and we hope that it become extra-extraordinary.
"We're sat at a table where we've not been used to sitting: now we want to eat as well."
Juventus have 74 points from 34 games and AC Milan 71 with
Milan laboured through the first half and even though they upped the pace in the second they still struggled to break down relegation candidates Genoa.
The breakthrough came four minutes from full time however as Urby Emanuelson outmuscled Davide Biondini for the ball in midfield, and set off on a driving run down the left. His cross was helped on to the far post by Zlatan Ibrahimovic, where it found second-half substitute Boateng who fired the ball under Genoa goalkeeper Sebastien Frey.
The game pitted the division's best attack (Milan with 63 goals scored) against its worse (Genoa with 62 conceded), but neither lived up to its reputation in the first half which offered nothing in the way of chances or meaningful action.
Milan did have a brief sniff of goal when Emanuelson's run from halfway brought Stephan El Shaarawy into play, but his near-post cross for Ibrahimovic was cut out. The Swede also went down under a weak challenge in search of a penalty as Milan were reduced to trying to bag a spot-kick with nothing forthcoming through open play.
The second half saw an improved effort as Ibrahimovic forced a save from Frey, and Emanuelson volleyed just wide. The introduction of Boateng just after half time was a major influence on the game, as was Bosko Jankovic's sending off in the 72nd minute for a foolish trip on Ignazio Abate which earned him a second booking.
Antonio Cassano, who came on with Boateng, curled an effort just wide as time crept by, and when it seemed that even an exerted effort from Milan would not produce any reward Boateng popped up with a crucial goal to seal a valuable three points.