That means the two sides will do it all again at the Emirates Stadium a week on Wednesday, with the winner earning a fourth-round trip to Brighton.
The fixture means Arsenal could face a further six games this month, including their rescheduled Premier League match with West Ham.
Swansea, who meet Chelsea in the first leg of the League Cup semi-finals on Wednesday, would end up playing seven more games this month if they were to emerge triumphant in the replay.
Laudrup said: "The last thing I want is to lose, but to be honest a replay is not exactly what I am looking forward to with the two Chelsea games in mind and other league games.
"It is very tight, but when you have been at this profession at the highest level you always want to play and win."
Wenger added: "I am frustrated as I did not want a replay, but if it is the choice between staying in or going out then I would rather take the replay at the Emirates, they have a lot of games coming up too."
The tie had meandered along with little incident of note during a tepid opening half, but it sprung to life in the second 45 minutes.
Michu, who settled last month's league meeting between the sides, had looked to again be Arsenal's downfall as he slotted his 15th goal of the season just 73 seconds after coming on as a second-half substitute.
But two goals in three minutes from Lukas Podolski and Kieran Gibbs put the Gunners on course to advance, only for Graham, who scored a late leveller against Aston Villa last week, to rifle home with three minutes left.