Fulham put six past Port Vale
Zamora, who could well feature in Fabio Capello's plans for England's Euro 2012 qualifiers against Bulgaria and Switzerland next week, scored in each half at Craven Cottage while Zoltan Gera also netted twice.
Newcastle United also overcame fourth tier opposition although they had some anxious moments in a 3-2 win at Accrington Stanley.
Everton completed a good night for the Premier League sides with a 5-1 home defeat of Huddersfield Town.
Blackpool were the only top flight side to fall in the second round, going down 4-3 to League One MK Dons on Tuesday.
Holders Manchester United and England's other clubs involved in the Champions League or Europa League enter the competition in round three.