Chasing 158 for a victory, Chennai rode on in-form opener Smith's 51-ball 57 and significant contributions from Faf de Plessis (31) and Dhoni (22 not out of 12 balls) to make 160 for six in 19.3 overs to end Mumbai's superb winning streak which went back to the beginning of last season.
Chennai skipper Dhoni struck a six and a four in the last over bowled by Kieron Pollard to guide the visitors home after they needed 17 off the last two overs and 11 in final over.
This was Chennai's seventh victory from ninth games and they caught up with Kings XI Punjab on 14 points but with one less game in hand.
After today's defeat, Mumbai, who had got their campaign back on track with back-to-back victories at this venue after five straight defeats in the UAE, went a step closer to elimination from the play-offs and they now need to win all their remaining games to avoid it.
Chennai commenced their run chase at a slightly slower rate as compared to the hosts, making 38 for 1 at the end of the first six overs of power play.
Smith started off in blazing fashion by hitting Praveen Kumar for two fours in the opening over.
Brendon McCullum (13) followed suit at the other end when he carted Jasprit Bumrah for a six over the straight field and a four with a cut shot, before playing across to Kumar to be trapped leg before the wicket.
It was a blow to Chennai as the opening wicket duo had been the fulcrum of their big scores this season.
Suresh Raina (19) looked in good nick though he was dropped at third man by substitute fielder Apoorv Wankhede off Bumraj when on five. However, the hosts did not have too long to wait for his dismissal as he was stumped off a wide ball down the leg from Pollard to leave Chennai at a tricky 56 for 2 in the eighth over.
Smith then upped the tempo by striking a four and a huge six over the straight field off his West Indian compatriot Pollard to help Chennai reach 75 for 2 after 10 overs.
The partnership of 63 in 46 balls between Smith and du Plessis was ended when Smith tried to repeat his six over mid wicket off Praveen Kumar and was caught in the deep. The dismissal of du Plessis, yorked by Lasith Malinga, 10 runs later at 129 raised Mumbai's hopes.
The hosts were a bit unlucky when Ravindra Jadeja escaped after a strong and confident shout for leg before off the first ball he faced from Malinga. But, he played onto the stumps against the same bowler later.
Mithun Manhas was run out going for a second run off the first ball he faced and Chennai were left to score 11 in the last over for a win and Dhoni achieved it by carting Pollard over the mid wicket region for a six and then cover-driving the West Indian to the fence for the winning runs.
Earlier, Ambati Rayudu notched up his first half century of the season as defending champions Mumbai battled their way to 157 for 6.
One-down batsman Rayudu, whose previous highest score this season was 48, made 59 with four sixes and two fours in 43 balls before he was out in the 19th over.
Rayudu's stand of 61 in 54 balls with opener Lendl Simmons (38), playing his first game after being inducted into the squad in place of injured Jalaj Saxena and replacing Ben Dunk in the eleven, laid the platform after the early loss of opener Chidambaram Gautam for 9 in the third over.
Towards the end, left hander Corey Anderson (18 not out in six balls with two sixes) and Aditya Tare (9 with two fours) put on 28 runs in 12 balls to give the total some respectability after Mumbai were asked to bat first.
Mumbai made a decent start despite losing opener Gautam cheaply. The home team scored on 42 for 1 in the powerplay period of six overs. Simmons carted fellow West Indian Samuel Badree for two sixes over in the sixth over.
But, Mumbai scoring rate slumped later as they could add only 12 runs in the next three overs. The hosts accounted for 33 dot balls in the first nine overs at the end of which the score read 54 for 1. It became 57 for 1 after 10 overs.
Rayudu escaped being caught when Mithun Manhas made a mess of a chance at deep extra cover off Ravindra Jadeja and then celebrated it by slog-sweeping the left-arm spinner for a six over midwicket.
Simmons departed when he tried to give charge to R Ashwin and was caught at long on for a run-a-ball 38 that included two sixes and three fours. The stand between him and Rayudu was worth 61 off 54 balls.
The next five overs produced 42 runs leaving Mumbai at 99 for 2 after 15 overs. Though Rayudu carted Jadeja for another six over long on, things had been kept tight by CSK bowlers.