Opener Morgan's 69-run stand with Jacques Kallis (45) took Kolkata within touching distance of their 157-run target, made possible earlier by the Kochi skipper Mahela Jayawardene (55) and Brad Hodge (35).
But Kochi staged a late turnaround thanks to Raiphi Gomez's double strike and Vinay Kumar's two for 28 to restrict Kolkata to 139 for seven.
Jaidev Unadkat's tight early spell proved profitable for Kolkata, who won the toss, as Brendon McCullum fell in the second over and Parthiv Patel departed after his promising 21.
The Tuskers trudged to 38 in seven overs, before Jayawardene cut loose in a 51-run stand with the opener Michael Klinger.
The Australian's exit for 29 was followed by Ravindra Jadeja's (eight), but Hodge powered his way to 35 in just 19 balls.
Jayawardene reached 50 in 38 balls before being run out in the 18th over, while Hodge helped Kochi finish on a high by smashing Brett Lee for 21.
Kolkata's platform came via Morgan and Kallis, who made light work of Kochi's pace attack to score at around eight runs an over.
Kochi fought back as spinner Gomez struck twice in the 11th over to remove Kallis and captain Gautam Gambhir first ball, but Morgan then took charge of proceedings.
With 57 needed off the last five overs, the left-hander overcame the loss of Manoj Tiwary to reach 50 and began the late assault.
But a mix-up with Yusuf Pathan (17) left him run-out in the penultimate over, with Pathan himself caught next ball, while Brett Lee and Mark Boucher were dismissed in the final over.
* Compiled by The National staff with agencies