The first half-chance of the match fell to Ireland inside four minutes, but Keith Andrews's effort drifted high and wide from just inside the box.
The away side began to dominate possession as the half approached and Thomas Muller twice came close with headed efforts either side of the 20-minute mark.
Germany continued to pass and probe and Sami Khedira was next to come close just before the 30-minute mark, but his effort was blocked by O'Shea.
Ireland continued to struggle for invention late in the opening period and Reus scored his second five minutes before the break after good work from Jerome Boateng.
Germany were not finished there and Klose made it four after 58 minutes when he converted from a tight angle to pile more misery on the home side. Meanwhile, it was five just past the hour mark when Toni Kroos finished from close range.
The home side continued to struggle late in the match, with an Andrews effort after 75 minutes sailing past the post from outside the box.
Germany continued to threaten in the latter stages and Kroos added their sixth with a cool finish five minutes from time. Meanwhile, Ireland grabbed a late consolation via Andrew Keogh to give their supporters something to cheer about.