Long scored the winner in the 78th minute after being picked out by winger Aiden McGeady and the home side nearly doubled the lead with a succession of shots by Jon Walters in the closing minutes.
"I think we deserved to win because the possession [we had in] the first and also the second half," coach Giovanni Trapattoni told reporters.
"I think we could also have scored two. [We had] three or four good opportunities."
Trapattoni learned after the game that midfielder Keith Fahey would miss Euro 2012 after aggravating a groin strain in the warm-up.
"He was scanned following the match," a spokesperson from the Football Association of Ireland told the Irish Times. "The results [of the scan] were discussed with the player and he agreed that he would not recover fully in time for Euro 2012."
Paul Green, who is unattached after leaving Derby County, has been called up to replace Birmingham City's Fahey.
Bosnia, who missed out on qualification for Euro 2012 following a play-off defeat by Portugal, failed to capitalise on a string of chances against a weakened Irish defence missing defender John O'Shea and veteran goalkeeper Shay Given.
Ireland, who qualified for their first major tournament in a decade by beating Estonia, produced a fine attacking display that belied their reputation as a very defensive team.
They face world and European champions Spain next month in a tough Group C in Ukraine and Poland that also includes Italy and Croatia. They play Hungary on June 4 in their final friendly before the tournament.
McGeady's man-of-the-match winning performance and an excellent display by Sunderland's McClean on his first start for Ireland may give Trapattoni a selection dilemma on the flanks.
McClean, cheered by the crowd every time he got the ball, made his case for a start ahead of Damien Duff with incisive runs and a shot well saved in the 41st minute.
"Competition is of course the life of the team," Trapattoni said when asked about the performance of players not normally in the first team, adding that McClean faded in the second half.
Ireland struggled to cope with a physical Bosnian team in the first 20 minutes, with striker Edin Dzeko threading the ball across an open goal mouth but Vedad Ibisevic failed to finish.
Ireland improved later in the half but Robbie Keane, Darron Gibson and Duff all failed to capitalise on good opportunities.
Both sides had chances in an open second half with McGeady and Walters hitting the woodwork and Bosnia's Miroslav Stevanovic rolling the ball across the face of the Irish goal.
The team treated the crowd of 37,100 to a lap of honour ahead of their flight to Italy on Sunday for a pre-tournament training camp.
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