Martin O'Neill's team had the advantage of an away goal from their first leg 1-1 draw in Zenica, but they wasted little time in building on that when Walters converted a first-half penalty.
The Stoke City man missed the first leg due to suspension but returned to the starting line-up and kept his cool on 24 minutes when Ireland were awarded a spot-kick.
Walters got his second of the game on 70 minutes when firing in at the back post following a superb cross from Robbie Brady, who was awarded man of the match.
Ireland played a positive brand of football and kept their opponents quiet for the most part, although goalkeeper Darren Randolph was on high alert to deal with any danger.
The visitors did threaten several times and almost pulled a goal back late on but Ireland held on to record a historic victory that takes them to the Euro finals in France next summer.
After finishing third in qualifying, Ireland made it through to the Play-Off and they certainly made the most of that by overcoming a very difficult opponent to join Europe's elite on the biggest stage.
Republic of Ireland: Randolph; Coleman, Keogh, Clark, Brady; Hendrick, Whelan (O'Shea 90), McCarthy, Walters; Hoolahan (McClean 55); Murphy (Long 55).
Bosnia & Herzegovina: Begovic; Cocalic (Besic 46), Spahic, Kolasinac, Vranjes, Medunjanin (Djuric 69), Pjanic, Lulic (Ibisevic 80), Zukanovic, Visca, Dzeko.
Referee: Bjorn Kuipers (Netherlands).