The Gunners were branded "timid" by manager Arsene Wenger following Sunday's 2-0 Premier League defeat by Manchester City - a result that extends their winless streak in 2013 to three matches.
Wenger's men fought back to lead 2-1 in the original third round tie with Swansea before a late Danny Graham goal sent the game to a replay.
Vermaelen, 27, was an unused substitute during that match but saw enough to be confident ahead of the replay at Emirates Stadium.
"Looking at the Swansea game in its entirety, I think we deserved to win. We had plenty of chances and I don't think anybody could have complained if we had gone through to the next round," Vermaelen said.
"At 2-1 up that late in the game, you would like to see it through. It is never nice to concede in the final few minutes, but we will look forward to the replay and feel that our FA Cup destiny lies in our own hands."
Wenger felt his men paid the price for not closing out the original tie after they
had recovered from going behind to Michu's strike just before the hour thanks to goals from Lukas Podolski and Kieran Gibbs.
"After going 2-1 up I felt we were a bit too nervous in some situations and
lacked defensive serenity," said the Frenchman. "Hopefully this will change and we will gain some more confidence defensively."
Arsenal will be without defender Laurent Koscielny after his red card against Man City and injured midfielder Mikel Arteta as they look to set up a fourth-round clash at Championship side Brighton.