De Villiers hit a fighting 66 and shared a match-winning 113-run stand with Mandeep Singh (54 not out) as Bangalore recovered from a sluggish start to post a challenging 180 for four after electing to bat.
Rajasthan had the proceedings firmly under their grip but they leaked 86 runs in the last six overs to let it slip.
De Villiers' 66 came off 38 balls with four sixes and as many fours while Mandeep took 34 balls for his unbeaten 54, which had seven fours and two sixes.
Rajasthan's surprise capitulation reduced it to a lop-sided affair and their total was the lowest in the IPL Play-offs.
Bangalore will now play against Chennai Sper Kings in the second Qualifier on Friday, the winner of which will lock horns with Mumbai Indians in the summit clash on Sunday.
Rajasthan lost in-form Shane Watson (10) and Sanju Samson (5) cheaply and the onus was on Rahane and skipper Steve Smith to bring the side back in the contest.
The required rate had zoomed up to 15 and Rahane was left with no option but to go all out and in the process he lost his wicket when he swept Yuzvendra Chahal, managed only an edge, which was easily taken by De Villiers.
Nothing was going Rajasthan's way as James Faulkner was dismissed in a bizzare fashion. The Australian hit a low full toss from Sreenath Aravind, the edge hit the toe of the batsman and flew towards the bowler, who took an easy catch in his follow through.
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