The Juve winger struck just before half-time to cement the Old Lady's position at the top of the Italian Serie A table.
Furthermore, it gives Antonio Conte's men confidence for their next two meetings with La Viola in the last 16 of the UEFA Europa League.
Jose Callejon scored the only goal of the game nine minutes from time as Rafa Benitez's men strengthened their claim for UEFA Champions League football against next season.
Palacio nodded in the only goal of the game on the half-hour mark as Inter returned to winning ways after draws in their previous two league games.
Maximiliano Moralez's goal on the hour mark was the difference between the two sides at the Stadio Olimpico.
Lazio shot themselves in the foot two minutes after conceding the goal when playmaker Antonio Candreva was sent off for a second yellow card.
Paloschi put the hosts ahead in the fifth minute and that goal looked as if it would be the winner for the much the match.
However, in a dramatic finale, Alberto Gilardino equalised two minutes from time for Genoa, only for Paloschi to convert his second spot-kick on the final play of the game after substitute Dejan Lazarevic was fouled by Marco Motta in the area.
Ibrahima Mbaye scored his first goals in Serie A to crown one of Livorno's best halves of football this season, but that was still not enough to ease their relegation worries.
Instead, Domenico Di Carlo's men capitulated in the second half with Nenad Krsticic bringing Samp back into the game before a Federico Ceccherini own goal drew them level.
Stefano Okaka and Manolo Gabbiadini then sealed the win for the Blucerchiati to take them 13 points clear of their opponents, who remain inside the bottom three having wasted a great chance to move out of danger.