Berezovski was red-carded in the 26th minute for handling an attempted lob by Ireland striker Simon Cox outside his area, although television replays suggested the ball hit the goalkeeper's chest.
Ireland took full advantage, going ahead in the 42nd through Valeri Aleksanyan's own goal before center back Richard Dunne added a second in the 59th.
Henrik Mkhitaryan replied two minutes later for Armenia, which would have finished runner-up in Group B with a win in Dublin, before Ireland striker Kevin Doyle was shown a second yellow card in the 80th.
Armenia was attempting to reach the finals of a major tournament for the first time but its hopes virtually disappeared when Spanish referee Eduardo Iturralde Gonzalez sent Berezovski off.
In addition to the fact that Berezovski may not have touched the ball with his arm, Gonzalez should have whistled moments earlier because Cox had controlled the long through ball from Glenn Whelan with his outstretched right arm.
Ireland was on the wrong end of a referee's handball decision two years ago in a qualification playoff against France for the 2010 World Cup.
Ireland also finished with 10 men after Doyle was dismissed for elbowing his marker as he challenged for a high ball, but the damage had been done by then for Armenia.
Aleksanyan's own goal was completely avoidable, the defender glancing the ball into his own net from five yards (meters) without an Irishman near him after Doyle had failed to get anything on Damien Duff's cross.
Dunne then scored his first international goal in two years, bundling in a cross from Aiden McGeady that was flapped at by replacement goalkeeper Arsen Petrosyan.
Armenia was given a flicker of hope two minutes later when Ireland goalkeeper Shay Given failed to keep out Mkhitaryan's curling shot from edge of area.
However, Ireland survived without any further trouble and awaits its playoff opponent in Thursday's draw.
Ireland: Shay Given, John O'Shea, Sean St. Ledger, Richard Dunne, Stephen Kelly, Damien Duff, Keith Andrews, Glenn Whelan (Keith Fahey, 76), Aiden McGeady (Stephen Hunt, 67), Kevin Doyle, Simon Cox (Jon Walters, 80).
Armenia: Roman Berezovski, Sargis Hovsepyan, Karlen Mkrtchyan, Marcos Pizzelli (Edgar Manucharyan, 62), Edgar Malakyan (Arsen Petrosyan, 28), Gevorg Ghazaryan (Artur Sarkisov, 62), Yura Movsisyan, Hrayr Mkoyan, Valeri Aleksanyan, Henrik Mkhitaryan, Levon Hayrapetyan.