Skipper Tendulkar hammered an unbeaten 100 off just 66 balls for his maiden IPL hundred to help his side reach 182-2 before Kochi achieved the stiff target with six balls to spare for their first win in three matches.
New Zealander McCullum cracked a 60-ball 81 while Sri Lankan Mahela Jayawardene hit 56 off 36 balls before being yorked by compatriot Lasith Malinga, but not before the pair had put on 128 for the opening wicket.
McCullum hit two sixes and 10 fours and Jayawardene nine fours.
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Ravindra Jadeja (25 not out) finished the match with two successive sixes off left-arm spinner Ali Murtaza as Mumbai suffered their first defeat in three games.
Tendulkar, needing 10 runs off the last four deliveries of the innings to complete the century on his home ground in Mumbai, hit two fours and took two singles off seamer Vinay Kumar.
The Mumbai captain, who smashed three sixes and 12 fours, added 116 for the second wicket with Ambati Rayudu, who hit four sixes in a 33-ball 53.