Zeman, who led Pescara to promotion to Serie A in 2012 before joining Roma, returned to the club this week following the sacking of Massimo Oddo.
His impact was immediate as Serie A's bottom club raced into a 3-0 lead before half time and added the gloss to a handsome win in the second.
Roma retook second spot with a 4-1 win over Torino, who had Joe Hart in goal.
Former Manchester City striker Edin Dzeko and ex-Chelsea midfielder Mohamed Salah both scored past the on-loan Manchester City keeper.
Napoli had briefly moved into second with a 3-1 win at Chievo, earlier on Sunday.
After Lorenzo Insigne had given them the lead, Marek Hamsik scored to move within three goals of equalling Diego Maradona's record tally of 115 for the club.
Piotr Zielinski made it 3-0 before Riccardo Meggiorini scored a consolation for the home side.
Roma are seven points behind leaders Juventus, and have two more than Napoli.
The story of the day, though, belonged to Pescara and their returning coach.
Czech Zeman, 69, claimed this week that he "owed" the club something having left them for the second of his two spells at Roma four and a half years ago.
Genoa's Lucas Orban deflected an Alberto Cerri shot in to his own net to give Pescara the lead before Gianluca Caprari and Ahmad Benali put the home side in complete control.
Caprari got his second in the 81st minute before Cerri sealed the rout in the closing stages.
It was only Pescara's second win of the season and their first since a 3-0 success at Sassuolo.
They remain rooted to the bottom on 12 points, 10 adrift of fourth-from-bottom Empoli.
Elsewhere, Gabriel Barbosa's 81st-minute goal gave Inter Milan a 1-0 win at Bologna, Gregoire Defrel scored twice as Sassuolo came from behind to win at Udinese, while Fabio Quagliarella cancelled out Mauricio Isla's opener as Sampdoria drew 1-1 with Cagliari.
New coach Zdenek Zeman led Pescara to their first win since August - a stunning 5-0 victory over Genoa.