The midfielder, who is on loan from QPR, put Milan ahead in the 20th minute before setting up Keisuke Honda in the second half.
Marco Motta reduced the deficit in the 73rd minute but the hosts could not find an equaliser as Milan claimed their third successive Serie A victory - the first time that has happened all season.
With six games remaining, the Rossoneri are five points adrift of city rivals Inter Milan, who hold the last Europa League qualifying spot.
Genoa, however, lost for the second week in a row and are six points adrift of Milan.
The visitors left Mario Balotelli out of the starting line-up because of illness and Genoa dominated possession early on, but the visitors took the lead following a surging run from Taarabt.
The former Fulham player drilled a right-footed shot towards the far post and out of Genoa goalkeeper Mattia Perin's reach.
At the other end, Stefano Sturaro got past his marker only to see his 30-yard effort go wide.
Genoa continued to press forward and, in the 37th minute, Andrea Bertolacci's left-footed attempt towards the far post went wide of Milan goalkeeper Christian Abbiati's far post.
Milan's Philippe Mexes almost scored an own goal before the interval after he attempted to clear a dangerous cross. Fortunately for Clarence Seedorf's side, Mexes headed the ball just over the crossbar.
Genoa showed more determination after the re-start and, in the 52nd minute, Ioannis Fetfatzidis tested Abbiati with a hard drive that the keeper saved.
Genoa almost pulled level when Sebastian De Maio headed Stefano Sturaro's cross inches wide.
Milan then doubled their lead in the 56th minute against the run of play. Taarabt's fantastic ball found Honda outside the area and he smashed a left-footed shot past Perin.
Genoa reduced the arrears when Fetfatzidis's cross fell to Motta and he struck the ball from the heart of the area past Abbiati.
Two minutes later, Abbiati had to dive in order to clear Sturaro's goal-bound attempt, but Genoa eventually ran out of steam as Milan held on for victory.