Set only 140 to win in the third match of the series, which also features Pakistan, Sri Lanka were wobbling at 120 for five.
But with 18 runs needed from the last 24 balls, Dishan struck four boundaries as Sri Lanka won with 14 balls to spare.
New Zealand and Pakistan head the table with six points followed by Sri Lanka on five after the first round.
The New Zealanders managed only 139 runs from 43.4 overs with Muthiah Muralitharan taking three for 16 from 8.1 overs.
Prabath Nissanka created the early problems, taking two for 11 from five fiery overs with the new ball.
New Zealand, who lost the toss, slumped to 18 for three in the sixth over after two acrobatic catches from Kumar Sangakkara in the slips and a run out.
Number three batsman Lou Vincent top scored with 32 from 72 balls hitting two boundaries.
Vincent helped avoid an embarrassingly small total by sharing a 48-run stand for the fifth wicket with left-hander Jacob Oram (20).
Muralitharan eventually broke through as Oram popped back a return catch and Sri Lanka wrapped up the innings smartly with the last six wickets falling for 45 runs.
Opener Romesh Kaluwitharana top scored for Sri Lanka with 48 from 119 balls, including four fours.
He put on 68 for the first wicket with Sanath Jayasuriya, who battled against tonsillitis during his 33.
After Oram dismissed Jayasuriya and Kumar Sangakkara, who trod on his stumps, Kaluwitharana added 47 runs with his skipper Marvan Atapattu.
Oram struck again, dismissing Kaluwitharana and captain Marvan Atapattu in quick succession, before Daryl Tuffey had Mahela Jayawardene caught behind.
Dilshan then came to the rescue although Sri Lankas missed out on a valuable bonus point by failing to win within 36.4 overs.
The match was reduced to 46 overs after morning drizzle. The final four matches of the tournament have been shifted to Dambulla after the early arrival of the southwest monsoon in Colombo.
Scoreboard
New Zealand
C.Nevin run out (Atapattu) 12
S.Fleming c Sangakkara b Nissanka 2
L.Vincent lbw Jayasuriya 32
S.Styris c Sangakkara b Nissanka 0
C.Cairns c and b Gamage 14
J.Oram c and b Muralitharan 20
C.Harris not out 20
B.McCullum lbw Muralitharan 0
K.Mills b Dharmasena 6
D.Vettori run out 5
D.Tuffey lbw Muralitharan 2
Extras (lb-12, w-9, nb-1, b-4) 26
Total (all out, 43.1 overs) 139
Fall of wickets: 1-6, 2-18, 3-18, 4-46, 5-94, 6-102, 7-108, 8-127, 9-135, 10-139.
Bowling: Vaas 9-2-23-0, Nissanka 5-0-11-2, Gamage 5-1-26-1, Dharmasena 8-0-28-1, Muralitharan 8.1-1-16-3, Jayasuriya 8-1-19-1.
Sri Lanka
S.Jayasuriya c Vettori b Oram 33
R.Kaluwitharana c McCullum b Oram 48
K.Sangakkara hit wicket b Oram 1
M.Atapattu c Vincent b Styris 18
M.Jayawardene c McCullum b Tuffey 2
T.Dilshan not out 18
C.Vaas not out 5
Extras (lb-8, w-2, nb-6, b-2) 18
Total (five wickets, 43.4 overs) 143
Fall of wickets: 1-68, 2-69, 3-116, 4-116, 5-120
Did not bat: Prabath Nissanka, Muttiah Muralitharan, Dharshana Gamage, Kumar Dharmasena
Bowling: Tuffey 7-0-37-1 (nb-1), Mills 5-0-32-0 (nb-5), Oram 9-4-12-3 (w-1), Vettori 9-1-19-0, Harris 10-0-27-0 (w-1), Styris 3.4-2-6-1.
Result: Sri Lanka won by five wickets
