The Gunners were lambasted as "timid" by manager Arsene Wenger after going down 2-0 at home to Manchester City as their Premier League ambitions suffered another setback on Sunday.
However, with a fourth-round tie at Championship side Brighton awaiting the winners of Wednesday's tie, hopes of ending an eight-year trophy drought could well be rekindled if Arsenal can find their form against the Swans.
Vermaelen was an unused substitute at the Liberty Stadium last weekend, but the Belgian centre-back saw enough from his side's showing in the 2-2 draw - when they had fought back well to lead going into the final three minutes - to be confident of a positive conclusion.
"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."
Swansea could yet be heading to Wembley in the Capital One Cup, having beaten Chelsea 2-0 at Stamford Bridge in the first-leg of their semi-final. In the other tie, League Two side Bradford - who had knocked out Arsenal on penalties in the previous round - also pulled off a shock 3-1 home win over Aston Villa.
Vermaelen feels teams can no longer view any matches as walkovers.
"Last week's results are just further evidence that the level in England is constantly rising," Vermaelen added in the official Arsenal matchday magazine.
"The lower teams, if you can call them that any more, are drawing closer to the top.
"They all have strong, athletic players who give you a tough time.
"There are not any easy games in the Premier League nowadays. The standard is going up and up, whomever you are playing against."
With a trip to Chelsea on Sunday, Arsenal could certainly have done without the replay, where they will be without defender Laurent Koscielny after his red card against City and injured midfielder Mikel Arteta.
Wenger felt his men paid a heavy price for not closing out the tie after they had recovered from going behind to Michu's goal just before the hour.
"After going 2-1 up I felt we were a bit too nervous in some situations and lacked defensive serenity," he said.
"Hopefully this will change and we will gain some more confidence defensively."
One player certainly not short of that at the moment is England full-back Kieran Gibbs.
The 23-year-old has produced a string of consistent performances and displayed his attacking flair with a superb volley which had put Arsenal in front late on with a second goal in the space of two minutes.
"When he plays with freedom and determination, you see the kind of performance he produced at Swansea," Wenger said.
"Gibbs is becoming a better player every year because he thinks about things and considers how he can improve.
"He can dribble, and you don't see many left-backs who can do that, he can also pass people, link well with other players, crosses well.
"If he is starting to finish too then that shows you the variety in his game."
Much has been made of Arsenal's failure to win a trophy since moving from Highbury to the Emirates Stadium.
However, former Gunners striker Dennis Bergkamp is in no doubt Wenger's current crop of homegrown players such as Gibbs and Jack Wilshere can soon have their time.
The Dutchman - who won the FA Cup four times and also three Premier League titles under Wenger - told FATV: "It is fantastic for Arsenal, but it's fantastic for English football that you can keep those players in your own country and playing in the first team.
"There was a complaint all the time when I played in England that, for all of the teams, there were too many foreigners and it kept back the young players from getting through the system to play in the first teams.
"But this is a different thing as you've got an English club with a lot of young English talent committing themselves to a club.
"It's an ideal situation so it's just a matter of getting them in the right place and you've got a fantastic future there."