Mumbai scored 164-5 in their innings, their top order once again failing.
Aaron Finch suffered a hamstring injury and was forced to retire hurt and they slumped to 45-3.
They were bailed out by West Indies star Kieron Pollard (70 off 34 balls) and New Zealand powerhouse Corey Anderson (50 off 38).
By the 14th over they were going at a little over five an over but from 83-5 in the 15th over they let loose, coming within two runs of doubling their total in their final five overs.
Kiwi Tim Southee took some serious punishment ??? his last two overs cost 36, which included four fours and three sixes.
The 19th over went from James Faulkner went for 18, and the last, from Chris Morris, for 15, as Mumbai set a decent total on what was not the easiest track.
Ajinkya Rahane (46) and Smith got the Royals up to 92-2 in the 14th over with some steady scoring and when the opener fell, the Royals needed 73 from 41 balls.
Deepak Hooda faced just four balls but hit two of them for six to give the Royals some momentum, Lasith Malinga removing his leg stump.
But Smith took 15 of a Malinga over, the 17th, leaving them with a target of 28 off 14, and they got home with five balls to spare. ??