Given excelled on his return from a knee injury before making a half-time exit, while O'Shea was given the full game to prepare for Sunday's opening Group C fixture against Croatia in Poznan.
With Croatia boss Slaven Bilic among the crowd, Ireland once again demonstrated the resilience which helped them qualify for the finals, although they failed to take any of the chances they created to win the game.
Defeat would have been harsh on Hungary, who controlled much of the game, and had it not been for the excellence of Given and replacement Keiren Westwood, as well as midfielder Stephen Hunt, they might have sent Giovanni Trapattoni's men on their way to Poland today on the back of a first defeat since March last year.
But the visitors survived to eclipse their 13-game unbeaten run under Brian Kerr between 2004 and 2006 and close in on Mick McCarthy's 16-game stretch more than a decade ago.
On a night when a clean bill of health was as important, if not more so, than the result, Ireland found themselves presented with a real test for a variety of reasons.
A spectacular thunderstorm over the Ferenc Puskas Stadium delayed the kick-off for 20 minutes and left the pitch slick, and once proceedings got under way Ireland found their hosts in little mood to give them an easy ride.
"It's very hard because it's a week away from the biggest games of our lives and, as much as we don't want to lose, we don't want to get injured and that all takes effect on the game," defender Richard Dunne said.
"We're happy because we've done our bit, we've got a bit of fitness out of it and we're still unbeaten going into the tournament."
Meanwhile, Germany midfielder Bastian Schweinsteiger shrugged off concerns over a thigh injury when he returned to training yesterday in Gdansk.
TONIGHT: Internationals: Sweden v Serbia (6.0), Turkey v Ukraine (7.0), Austria v Romania (7.30), France v Estonia (8.0).
Friendly: Glencairn v Ashfield (7.15).
LAST NIGHT: International: Hungary 0 Rep of Ireland 0.
Stagecoach Super Premier: Auchinleck 1 Irvine Meadow 2.
Super First: Shotts 3 Glenafton 2.