France required just a point to top qualifying Group D, but they appeared paralysed by nerves in the first half and fell behind shortly before half-time to a fine goal from Manchester City striker Edin Dzeko.
Already assured of a play-off place, Bosnia had played with nerveless intent but their resistance finally buckled in the 78th minute when Dzeko's City team-mate Nasri scored from the spot after he had been felled by Emir Spahic.
The win allowed France to avoid the trap they fell into in 1993, when a 3-2 defeat by Bulgaria denied them a place at the 1994 FIFA World Cup™, and means Blanc can continue his rebuilding work in Poland and Ukraine next summer.
"The match was difficult. It was what we expected," admitted Blanc. "But it's true that, even though we were poor for 45 minutes, Bosnia played very well. They were assured and showed good technical control. Their players caused us lots of problems. It was tough and it was tense. But in the end, it's a huge satisfaction to be qualified for the EURO."
Bosnia coach Safet Susic, whose side will learn their play-off opponents on Thursday, said: "I'm particularly sad for my players. I think they played a great match. They controlled the match for around 60 minutes before their physical level fell away. I hope we'll get more chance in the play-offs. I think my team deserves a place at EURO 2012."
Seeking to qualify for their first ever major tournament, Bosnia had pledged to take the game to France and they were true to their word in the first half. Careless in possession and inattentive in defence, France's players seemed cowed by the occasion and allowed Bosnia to put them under pressure repeatedly.
Senad Lulic found the side netting after charging down an Anthony Reveillere clearance, while Dzeko tested Hugo Lloris at his near post before blazing wide after being sent clean through by Miralem Pjanic.
Dzeko shocks home side
France's only opening of note saw Loic Remy over-run the ball with the goal in sight, but Bosnia were stroking the ball around with disregard for the hosts' anxiety and in the 40th minute they took the lead.
Sasa Papac caught the French defence napping with a quick free-kick to Dzeko, who rolled his marker and bent an exquisite shot inside Lloris' left-hand post from the edge of the penalty area.
Florent Malouda announced France's second-half intent by marauding down the left flank and firing a cross across goal, but there was no change in their fortunes and Blanc responded by introducing Kevin Gameiro and Marvin Martin.
Martin drew a save from substitute Bosnia goalkeeper Asmir Begovic with a low free-kick before Nasri almost went one better from similar range, only for his set-piece to be touched onto the crossbar by the Stoke City keeper.
Nasri finally nets
The pressure was mounting on the Bosnian goal and with 12 minutes remaining it finally told. Nasri showed dogged determination to rob Spahic inside the Bosnia box and when the former Montpellier defender brought him down, Scottish referee Craig Thomson had no hesitation in pointing to the spot.
The City midfielder made no mistake from 12 yards, slamming a crisp strike into the bottom-left corner to a roar of delight and relief from the home support. The hosts even came close to a winner in the last five minutes, but Gameiro could not get a touch to Jeremy Menez's low cross from the left. Fortunately for France, it mattered not.