Fulham, Leicester and Crystal Palace were among the other top-flight sides to progress.
Everton have suffered a disastrous start to their final season at Goodison Park, conceding seven goals in losing their opening two Premier League games without reply.
However, new signing Iliman Ndiaye gave the Toffees' support some cause for optimism by marking his full debut with a fine solo run and finish.
Tim Iroegbunam had also opened his Everton account by breaking the deadlock on 53 minutes before Beto added a third after Ndiaye's effort.
Brighton's victory came at a cost as new £25 million signing Matt O'Riley hobbled off less than 10 minutes into his debut after a wild challenge from Jay Williams that drew an apology from Crawley boss Scott Lindsey.
The League One side were outclassed at the Amex as Simon Adingra, Jeremy Sarmiento, Adam Webster and Mark O'Mahony struck for the Seagulls.
Palace put their disappointing start to the Premier League campaign aside to ease past Norwich 4-0.
Jean-Philippe Mateta scored twice for the Eagles with Daichi Kamada and Eberechi Eze also on target.
Leicester were also 4-0 winners over Tranmere as Jordan Ayew marked his full debut with a goal and assist.
A much-changed Fulham won 2-0 at Birmingham thanks to goals from Raul Jimenez and Jay Stansfield.
League Two Barrow provided the upset of the night by beating second-tier Derby on penalties after a 0-0 draw.
Barnsley, Blackpool, Bolton, Coventry, Fleetwood, Leyton Orient, Preston, Sheffield Wednesday, QPR, Stoke, Walsall and Watford were the other sides to book their places in round three.
Wednesday's action includes the only two all-Premier League ties as Newcastle visit Nottingham Forest and West Ham entertain Bournemouth.
The seven English sides to have qualified for European competitions receive a bye into the third round.
