The Boys in Green finished second in Group D and qualified for the playoffs by virtue of being one of the best eight runners-up in Europe.
Asked how big an achievement Monday's result is, Aldridge told Sky Sports News: "It's huge. I'm not getting too carried away because there's a lot of work to be done.
"I have to say, a few weeks ago I thought it was beyond us really. We managed to give ourselves a lifeline. It's brilliant, the way it happened. That feeling was magic.
"Martin [O'Neill] got his tactics bang on. Wales had some very good players, a good team. I thought it was a terrific all-round fighting display."
O'Ireland will find out their playoff opponents next Tuesday, with both ties taking place in the first fortnight of November.