Given excelled on his return from a knee injury before making a half-time exit, while O’Shea was given the full 90 minutes to prepare him 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 tomorrow 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.
Trapattoni wanted both Given and O’Shea to have a good workout in their first game since the end of the Barclays Premier League season, and Given’s health in particular was put to the test during the opening 45 minutes.
Dinamo Moscow midfielder Balazs Dzsudzsak was the main source of employment for the Aston Villa goalkeeper, first testing him with a skidding 17th-minute drive and then forcing him to make a fine save seconds later.
Given had already shifted his weight onto his right foot to cover another long-range effort, but after the ball took a wicked deflection he somehow managed to get back across and claw it away to safety one-handed.
His third save from Dzsudzsak did not have to be as spectacular, but he was kept on his toes throughout as Hungary, who finished third behind Holland and Sweden in qualification for the Euros, dominated for lengthy periods.
Ireland had started well and went close to the opening goal when O’Shea headed a seventh-minute Damien Duff cross onto the roof of the net.
Skipper Robbie Keane tested goalkeeper Adam Bogdan with a well-struck 14th-minute drive, but as Adam Pinter and Vladimir Koman got on top in the middle of the field, the home side began to make the running.
Richard Dunne made a perfectly-timed sliding tackle to deny Peter Szakaly a clear run on goal and Stephen Ward had to be equally resilient to dispossess striker Adam Szalai in the final minute of the half after he had worked his way into a shooting position inside the box.
The full-back needed treatment afterwards, but as Trapattoni held his breath, he returned to the field.
Given and striker Kevin Doyle did not after the break with Westwood and Jon Walters taking their place, but there was no suggestion that Given’s departure was the result of anything sinister.
Westwood had little time to settle as Hungary resumed where they had left off with Sean St Ledger and Glenn Whelan having to make important blocks, before the Sunderland goalkeeper proved his worth by getting down well to keep out another Dzsudzsak shot after it had been deflected towards goal by Szalai.
But as Trapattoni rang the changes – Keane, Duff and Keith Andrews all departed in quick succession as Simon Cox, Hunt and Darron Gibson were given a run-out – Ireland began to make an impression once again.
Walters forced a rare save from Bogdan and then drilled a right-foot shot across the face of goal, but Cox might have snatched the win with 12 minutes remaining.
Dunne’s towering defensive header set McGeady away and he played the ball into the path of the striker, whose shot was firm but did not elude the ever-vigilant Bogdan.
Westwood needed help six minutes from time when Imre Szabics’ effort looked to be heading straight for the bottom corner, but fellow substitute Hunt was perfectly placed to clear.