The home team took an early lead through Ryuki Kozawa in the third minute but it was neutralised by a strike from the visitors' Albert Zohmingmawia in the 18th.
Crossing over, Liberian Eric Brown scored in the 57th minute to put Mumbai 2-1 ahead before substitute Nabi deflected the ball into his own goal six minutes later to enable Aizawl draw level.
Aizawl are placed seventh with 9 points from 10 games.
Mumbai got off to a superb start by scoring from the penalty spot after Kozawa's attempted cross struck a defender's hand. The Japanese striker made no mistake by finding the left corner of the net.
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After trooping off the field at half time, Mumbai surged into the lead again off a free kick with Brown's shot going in off the near upright.
But the lead didn't last long. Zohmingmawia's high cross was deflected in by Nabi.
The visitors then missed a good chance when Joel Sunday blasted the ball over the bar in the 71st minute. Then Mumbai's Brown very nearly scored off another free-kick in the 88th minute but his power-packed long-range effort struck the post.