Hallgrimsson made three changes to the side beaten by England, one of which was enforced.
Seamus Coleman, who limped off against Lee Carsley's Three Lions, was replaced by Andrew Omobamidele, with Jason Knight and Alan Browne brought in for Adam Idah and Matt Doherty.
It was a tentative start to the game from the Republic, with a rather flat atmosphere accompanying the opening stages in a half empty Aviva.
The first chance arrived on 11 minutes as Konstantinos Koulierakis connected with Tasos Chatzigiovanis' looping corner, but his near-post header landed on the roof of the net.
After a sustained period of possession the Republic managed to fashion their first sight on goal with Will Smallbone slipping in Knight on the right-hand side of the area.
He squared the ball to Browne and his low strike was blocked before it could trouble Odysseas Vlachodimos.
Caoimhin Kelleher was called into action for the first time on 26 minutes, diving low to turn Tasos Bakasetas' curling free-kick around the post.
The Republic improved as the half wore on and Chiedozie Ogbene thought he had opened the scoring five minutes before the break.
He chested down a searching pass from Jayson Molumby and fired a powerful strike into the top corner before the flag went up for offside.
That effort raised spirits in the stands and on the pitch, with Browne guilty of squandering another good opportunity on the stroke of half-time.