The Republic pushed Group B winners Russia all the way throughout the campaign, but after Dick Advocaat's men won 1-0 in Slovakia last week, their hopes of automatic qualification were effectively over.
But that left them needing to secure at least a draw with surprise package Armenia at the Aviva Stadium on Tuesday night to book their play-off spot, and a 2-1 victory on a tense evening eventually saw them home.
Stoke midfielder Whelan said: "There was massive pressure on us because we knew if Armenia won the game, they could go through and nick that second spot.
"But we went out like in any other game trying to win the game and get the three points, and we did that.
"We are in the play-offs, but it's probably what we would expect, really. If we couldn't get the top [spot], we definitely wanted the play-offs.
"We are still in with a shout and you never know."
Ireland are seeded, along with Portugal, Croatia and Czech Republic, for Thursday lunchtime's draw in Krakow, and will meet either Turkey, Bosnia and Herzegovina, Estonia or Montenegro in the play-offs in November.
They could go into the first leg without both of manager Giovanni Trapattoni's first-choice strikers with skipper Robbie Keane battling a muscle injury and Kevin Doyle suspended after being sent off on Tuesday.
However, surprise selection West Brom frontman Simon Cox turned in a fine display against the Armenians, and with Shane Long and Jon Walters also waiting in the wings, the Italian has options.
Whelan said: "There were probably a few eyebrows raised beforehand, but he [Cox] came in and did a job.
"But there are other lads, not just himself, itching on the bench to come in and I am sure once they are given the jersey, one to 11, everyone is going to be trying their hardest."
Ireland's last brush with the play-offs ended in controversy when they went out to France in Paris two years ago when a missed handball decision ended their World Cup dreams.
But their fortunes improve dramatically on Tuesday when Armenia keeper Roman Berezovsky was harshly sent off for handling outside his area, while Valeri Aleksanyan's own goal and an error by substitute keeper Arsen Petrosyan, which led to Richard Dunne's second, helped to see Ireland through.
Midfielder Damien Duff said: "I don't know, maybe we were due a bit of luck.
"They say 'the luck of the Irish', but I don't think we have had it a lot over the years, so hopefully that can carry on next month into the play-offs."
Duff is one of the few members of the current squad to have played in the finals of a major tournament and he is desperate to repeat the experience.
However, asked about his preferences for the draw, he said: "I haven't even looked at it, to be fair. I just want to be in Poland and the Ukraine next summer."