Over 37,000 supporters filled the Aviva Stadium in Dublin on Saturday afternoon as the Irish played their last game on home soil ahead of the championship.
The goal came in the 78th minute when West Brom striker Shane Long headed home after numerous chances were denied by the post, bar and keeper Asmir Begovic.
It extended the Republic's unbeaten run to 13 games a day before they leave for their pre-tournament training camp in Italy.
And it ensures Giovanni Trapatonni's men - who face champions Spain, Croatia and Italy in a tough opening group - head off for Poland-Ukraine on the back of a win.
Asked how significant the win is, Long said: "It's massive.
"We joined up on Sunday, and we wanted to get off on a good foot today and get a bit of a roll going before we go to Poznan (Ireland's Euro 2012 base).
Job well done.
"We go on to Hungary on June 4 and we'll try to get a win there and get a bit of confidence going.
"The whole country's getting right behind us. They're all coming over to support us. Hopefully we'll give them something to shout about."
"The visitors held the ball and passed confidently for long periods with Manchester City striker Edin Dzeko keeping Richard Dunne on his toes throughout.
Miralem Pjanic and skipper Zvjezdan Misimovic ran the game from central midfield but the Republic were doggedly stubborn and Dunne made a series of important interventions.
In the process he proved his fitness after a mid-week fitness scare.
Meanwhile stand-in keeper Keiren Westwood, debutising for injured Shay Given, was called to make his first stop after 40 minutes.
The Republic gradually worked their way into the match after a strong start by Bosnia-Herzegovina.
McClean, starting a senior international for the first time, set up Darron Gibson to drag a 25th-minute effort wide after an enterprising foray inside from the left wing.
Skipper Robbie Keane sent a snapshot over the bar after turning smartly six minutes later.
Then McClean forced Begovic into a smart save at his near post with a well-struck left-foot drive, before the keeper had to block at the feet of Damien Duff after Keane's clever flick had put the midfielder in on goal.
The sides went into the break level but only the woodwork stopped a goal two minutes into the second half.
Trapattoni made a double change, replacing Glenn Whelan and Damien Duff with Keith Andrews and Aiden McGeady, and the latter almost made an instant impact.
He timed a perfect run to meet Kevin Doyle's knock-down and struck a right-foot volley goalwards - but with Begovic beaten, the ball came back off the post.
Walters very nearly gave the Republic a 74th-minute lead when he met McGeady's cross with a powerful header, but this time the crossbar denied the hosts.
However, Long finally made the pressure tell with 12 minutes remaining when he rose at the far post to head yet another inch-perfect McGeady delivery home.
It could have been 2-0 within a minute when McGeady put Walters in on goal, but club-mate Begovic closed him down well to block, and then pulled off a remarkable late save to deny Long a second.
"Today was good," said veteran defender Richard Dunne after full time. "When Aiden came on he changed the game and really opened them up.
"The atmosphere among us is really good. To win games obviously builds that.
"It's nice to win our final game in Dublin, to keep the crowd interested and excited for what's to come.
"We know against eastern European sides it's always hard.
"They're a tough side to play against, they're technically very good. But we've coped and we've kept a clean sheet and that's the big thing for us."