Needing 143 to win in what was a virtual quarter-final Giants ran out of overs as the Delhi unit fell two runs short of their target.
Lance Klusener, who was asked to bowl the last over of the match for Tigers, had to defend 14 runs off his last over. The South African started well, conceding just four runs off his first four deliveries and also picking up the wicket of T Sudhindra.
But a free hit off the fifth delivery was smashed to the boundary by Shane Bond which reduced the margin to three runs from one ball.
Klusener used all his experience to get away with just one run off the last ball.
For Tigers, it was a spirited effort from Man-of-the-Match Upul Chandana, who returned with the creditable figures of 3 for 14 in his quota of four overs.
He provided Tigers with much needed breakthroughs at crucial junctures to defend a modest total.
Shiv Sagar Singh, too, chipped in with a couple of wickets as no batsman, apart from the openers, Avishka Gunawardene (35 off 21 balls) and Mohnish Mishra (27 off 16 balls), could provide resistance to the Tigers' innings.
Mohnish Mishra, in particular, was looking dangerous as he hit five fours in 16 balls before captain Craig McMillan pulled off a splended catch to send him back.
Put in to bat first, Kolkata Tigers were off to a flying start with Deep Dasgupta leading the charge. He made a quick fire 35 off 21 balls which included four hits to the fence and one over it.
It took him just one over to adjust to the pitch and was particularly severe on Dhruv Mahajan whom he hit for 17 runs in his very first over.
However, the scoring rate dipped after Dasgupta's departure but Klusener held one end and played an uncharacteristic innings by his standards.
The former South African all-rounder hit eight well-timed boundries to score an unbeaten 62 off 61 balls in his 20 overs of stay at the wicket.
J P Yadav bowled a good line in the hot conditions and picked up the important wicket of Dasgupta who was looking dangerous for Giants. Shane Bond was generating a good pace as he too bowled economically but failed to pick up any wickets and conceded 15 runs of his 3.4 overs.
Kolkata Tigers 142/4 in 20 overs (Klusener 62, Dasgupta 35, McMillan 21; Ali Murtaza 1 for 26, JP Yadav 1 for 23, Dhruv Mahajan 1 for 37) beat Delhi Giants 141/8 in 20 (Gunawardene 35, Mohnish Mishra 27, T Sudhindra 14; Chandana 3 for 16, Shiv Sagar Singh 2 for 15).