Roma, who have the best goals total (26) and second worst defensive tally (20) in the league, led 2-0 at half-time through Francesco Totti and Pablo Osvaldo, a scoreline that was bound to test the frayed nerves of their supporters.
Zdenek Zeman's wildly unpredictable team, sixth in the table on 17 points, have twice squandered 2-0 leads and lost 3-2 at home this season but this time went on to add a third of their own through Erik Lamela in the 69th minute.
Destro came on in the 73rd minute. He was quickly booked, added the fourth goal six minutes later and was promptly sent off for taking his shirt off as he celebrated.
He will miss next week's derby against Lazio.
Josip Ilicic pulled one back for Palermo who are second from bottom with eight points.
Elsewhere, a blunder by defender Salvatore Aronica allowed Torino to score a stoppage-time equaliser to draw 1-1 at third-placed Napoli after the hosts had led for 85 minutes through Edinson Cavani's early strike.
The Torino goal, converted by Gianluca Sansone, ended Napoli's 100 percent home record and prevented them taking full advantage of leaders Juventus's 3-1 home defeat by Inter Milan on Saturday.
Forward Antonio Di Natale grabbed his fifth goal in three games to give Udinese a 1-1 draw at Bologna. Catania thrashed Lazio 4-0 helped by two goals from Alejandro Gomez while fourth-placed Fiorentina beat Cagliari 4-1.
Sampdoria slumped to a sixth consecutive defeat, losing 2-1 at home to Atalanta despite a brilliant bicycle-kick goal by Enzo Maresca.
Juventus lead the way with 28 points from 11 games followed by Inter on 27, Napoli on 23 and Fiorentina on 21.
CAVANI RETURNS
Uruguay forward Cavani returned for Napoli after missing two games through injury and made an immediate impact as he turned Marek Hamsik's pass into the net in the sixth minute.
On a wet afternoon at a half-empty San Paolo, sloppy Napoli wasted chances to seal victory especially when Hamsik rounded Jean Gillet and shot into the side-netting five minutes from time.
Aronica's lazy back-pass was then intercepted by Sansone who rounded Morgan De Sanctis to equalise for Torino.
Coach Walter Mazzarri missed the shambles leading up to Torino's equaliser as he had been sent off for protesting seconds earlier.
Di Natale, Serie A's top marksman in 2009/10, 2010/11 and joint third leading scorer last season, kept the goals flowing as his opportunist effort for Udinese wiped out Alessandro Diamanti's close-range goal for Bologna.
The 35-year-old Di Natale joined Cavani, Lamela and Inter Milan's Diego Milito on seven goals, one behind leading scorer Stephan El Shaaraway of AC Milan.
Second-half goals from Stevan Jovetic, Luca Toni and Jose Cuadrado gave Fiorentina their convincing win over Cagliari, handing visiting coach Ivo Pulga his first defeat with the Sardinians after winning his first four
