Tahir dismissed MI's top-order, forcing Parthiv Patel (19), Jos Buttler (38) and Rohit Sharma (3) back to pavilion in first seven overs.
Rahane hit six boundaries and three sixes in his 34-ball knock and shared 58 runs for the second wicket with Smith. Hardik Pandya then stunned the home crowd as Mumbai scored 30 runs from the final over as he scored four sixes and a boundary.
The team has retained many players from the last season and has bought Ben Stokes in the auction this year with 12 crores.
Smith had won the toss and elected to field against MI. Faf du Plessis missed out, while the hosts opted two spinners in Adam Zampa and Imran Tahir. But Rahane's dismissal meant Steven Smith took over the mantle of getting quick runs and he got stuck into the Mumbai attack. He was finally dismissed when Nitish Rana pulled off an excellent catch at deep square leg off the bowling of Tim Southee.
Smith's team-mate, England all-rounder Ben Stokes, contributed 21 in their 187-3. Earlier, opting to bowl after winning the toss, Mumbai Indian's leg-spinner Imran Tahir produced a sensational three-wicket burst in the space of 10 balls but Hardik Pandya's late heroics took MI to 184 for 8 in their opening match in IPL-10.
Mumbai Indians have given a look-in at Delhi young gun Nitish Rana, while Kieron Pollard has also been drafted in the playing XI despite joining the squad just a couple of days ago in Pune after flying in from the West Indies-Pakistan series.
Buttler was sent in to open the innings with Parthiv Patel, and that plan turned out to be a really good one, with the Englishman enjoying the ball coming onto the bat and, as a result, scoring at a rate of knots.
Ambati Rayudu's brief stay in the middle was ended by Rajat Bhatia, whose wily medium pace got the better of Rayudu and he gifted an easy return catch to Bhatia to depart for 10 as Mumbai Indians slumped to 92/4. But, at the fag end of the first session that saw fortunes swing like a pendulum, Hardik Pandya found a way to thump fast bowler Ashok Dinda for four mighty sixes for Mumbai Indians to set a challenging target of 185 for the Supergiant.