Claudio Ranieri's side are unbeaten in a club record-equalling 20 matches and lie six points behind Inter Milan who visit Napoli on Sunday.
Inter are top on 53 points while AC Milan, who beat Udinese 3-2 on Friday, are third on 45.
"It's clear we are behind and Inter are very strong. We'll do what we can and then we will see," midfielder Brighi told Sky television, aware Roma host the leaders at the end of March.
"We believe in ourselves because results breed confidence."
Palermo had the better of the early exchanges in Rome before Brighi's volley from a Francesco Totti corner was deflected in by defender Cesare Bovo just after the half hour.
Totti, a surprise inclusion after knee trouble, gave way to Julio Baptista at the break and the Brazilian soon netted his first goal of the league campaign when he tapped in Brighi's pass.
SUPERB STRIKE
The visitors protested about a push by Rodrigo Taddei in the build-up but could not complain about Bovo's woeful attempt to clear the ball for the second goal or Brighi's superb strike for the third.
Fabrizio Miccoli converted a consolation penalty for the Sicilians before John Arne Riise blasted Roma's fourth.
Striker Totti, considering coming out of international retirement for Italy at the World Cup, could be fit for Thursday's Europa League clash at Panathinaikos but will be suspended for next weekend's Serie A game against Catania.
Sampdoria moved above Napoli and Palermo into fourth spot after Franco Semioli and Giampaolo Pazzini headed in against their former side to secure a 2-0 home win over Fiorentina.
It was Sampdoria's fourth straight victory without the injured Antonio Cassano, who almost joined the Florence club last month.
Fiorentina, who have sunk to 11th after a poor run, lost Alessandro Gamberini and Mario Santana to injury in the first half and must pick themselves up for Wednesday's Champions League last 16 first-leg at Bayern Munich.
