The Irish picked up where they left off as Jason Knight drilled in a low effort that was easily held by Hradecky.
Ferguson showed superb feet to jink into the area but Rasmus Schuller nipped for a crucial intervention, and, from the resulting corner, Sammie Szmodics' flicked effort forced Bayern Leverkusen keeper Hradecky into another smart save.
They got their deserved equaliser when Brady's free-kick was well met by Scales, whose header flew into the corner of the net.
With the momentum in their favour, Ferguson fired low at Hradecky, but despite all the Irish pressure Finland should have taken the lead when the ball broke to Glen Kamara on the edge of the area but he went for power and it flew wide.
The in-form Troy Parrott, who has eight goals in 10 games for AZ Alkmaar, was introduced in place of Ferguson, but it was Finland who had another gilt-edged chance when Tomas Galvez dragged an effort wide from inside the area.
Finland could have snatched the game when Benjamin Kallman's header grazed the post with Kelleher stranded.
But the decisive moment came a minute later when substitute Ebosele danced his way through Galvez and Teemu Pukki before playing a brilliant cross to Brady at the back post, and the left back made no mistake to fire home.
It could have been even better in injury time when Parrott's effort was batted away by Hradecky and Adam Idah's follow up was straight at the goalkeeper.
It was the first time the Republic had come from behind to win a competitive match in 11 years, and they will look to make it back-to-back victories against Greece on Sunday.