Walters, back from suspension, converted a penalty in the first half before adding a second 20 minutes from time to send Ireland back to the European championships where they suffered a swift exit in 2012.
"It's amazing. Every single player's been part of this. Now we can put right what happened at the last European championships," Walters said in a pitchside interview.
Ireland's chances of even reaching the playoffs looked in peril midway through the campaign but the coaching duo of Martin O'Neill and Roy Keane, recruited at considerable expense, put some shape into a side that went on to record a shock victory over world champions Germany last month.
It may not have been especially pretty but even without the vast experience of John O'Shea and Shay Given, a makeshift partnership of Ciaran Clark and Richard Keogh defended manfully in front of Darren Randolph in both legs as Ireland managed the tie well.
O'Neill's side began the brighter and led after Daryl Murphy's cross struck Ervin Zukanovic on the arm and Walters slotted the harshly awarded penalty past his former Stoke City team mate Asmir Begovic to the delight of the raucous full house.
Bosnia, hoping to reach a second major tournament as an independent nation following last year's World Cup, began to find their feet late in the first half and that momentum continued after the break.
But they were poor throughout the two games and only had a snatched Senad Lulic chance to show for it as Walters arrived at the back post to coolly finish a deflected free kick from Robbie Brady, Ireland's scorer in the first leg.
Substitute Shane Long, the goalscoring hero against Germany, could have added a third but with Rudolph's goal untested until Vedad Ibisevic rattled the crossbar in injury time, the 'Ole, Ole, Oles' rang around the stadium as the clock ran down.
Ireland, unbeaten at home in the group stages, join fellow playoff winners Hungary at France 2016 while the final two spots will be decided on Tuesday when Sweden travel to Denmark and Ukraine play away to Slovenia, both defending first leg leads.
(Reporting by Padraic Halpin; editing by Justin Palmer and Ian Chadband)
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