"That could have been a potential banana skin tonight if we hadn't gone out and been professional," Cardiff boss Malky Mackay said.
Stoke progressed with a 3-1 victory against Walsall.
Luke Freeman put Stevenage ahead in the 36th minute and Gerard Deulofeu leveled just before half-time. Playing without England fullback Leighton Baines, Everton could have faced an early exit but substitute Greg Tansey missed a good chance in the last minute.
Nottingham Forest also advanced 2-1 after extra-time against Millwall, with Jamaal Lascelles heading home the winner.
The third-round draw produced a marquee fixture between Manchester United and Liverpool, to be played on Sept. 24 or 25. This game could see the return from suspension of Liverpool striker Luis Suarez, who was banned for biting Chelsea defender Branislav Ivanovic last season.
Other all-Premier League matches are: Aston Villa hosting Tottenham, Arsenal at West Bromwich Albion, West Ham playing Cardiff and Everton facing Fulham. Last season's winner Swansea will start the defense of its title against Birmingham and there will be a repeat of this year's FA Cup final, with holders Wigan travelling to Manchester City.