The seamer removed skipper Sachin Tendulkar and hard-hitting Kieron Pollard as Mumbai finished at 125-8 chasing a modest 136-run target to suffer their fourth defeat in 12 games of the Twenty20 tournament.
Bangalore made sure of a place in the semi-finals with 17 points. They are followed by Chennai (16), Mumbai (16), Kolkata (14), Rajasthan (11), Punjab (10), Kochi (10), Pune (eight), Deccan (eight) and Delhi (eight).
Mumbai needed 26 to win in the last over but managed only 15 as Anand dismissed West Indies all-rounder Pollard (24 off 13 balls) with his fourth delivery.
Tendulkar`s Mumbai were under pressure after being reduced to 25-3 in the opening 5.3 overs, with paceman Ishant Sharma taking two wickets.
Rajan, playing his first IPL match, then got two wickets in one over, including that of Tendulkar who top-scored with 37.
Mumbai paceman Dhawal Kulkarni grabbed three wickets to restrict Deccan, with Shikhar Dhawan (27 not out), skipper Kumar Sangakkara (27) and Sunny Sohal being the main scorers.
Mumbai seamer Munaf Patel, who conceded 23 runs in the last over, and Deccan batsman Amit Mishra exchanged words. Mishra hit the bowler for four successive boundaries in the over.
Deccan Chargers 135-6 in 20 overs (S. Dhawan 27 not out, K. Sangakkara 27, S. Sohal 20; D. Kulkarni 3-26) beat Mumbai Indians 125-8 in 20 overs (S. Tendulkar 37, K. Pollard 24; A. Rajan 3-27, I. Sharma 2-30) by 10 runs.
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