The Irish came into the game unbeaten at the top of the group following a 1-1 draw away to Denmark on Friday and, though they dominated possession throughout the match, they had a hard time hitting the target at the Aviva Stadium.
Striker David McGoldrick caused the visiting defence all sorts of problems and it was his 27th minute shot that Chipolina deflected into his own net to give Ireland the lead.
The home side created plenty of chances, with McGoldrick coming closest to doubling their advantage with a shot that hit the post before Brady stooped to head home in stoppage time.
Denmark had no such problems in Copenhagen, with Kasper Dolberg netting either side of a Christian Eriksen penalty and Yussuf Poulsen and substitute Martin Braithwaite also getting on the scoresheet in a resounding 5-1 win over Georgia.
Ireland have 10 points after four games with Denmark second on five from three. Switzerland, who finished fourth at the weekend's Nations League finals, are third on four points after two games with Georgia on three and Gibraltar bottom with none.
(Editing by Ken Ferris)
