Phillips hit 71 off 80 balls, Latham 68 off 74 and opener Young 54 off 64 balls to lift New Zealand from a precarious 110-4 after they were sent in to bat on a slow pitch.
Mark Chapman fired a rapid 25 off 12 balls with two fours and a six as New Zealand scored 62 runs in the last five overs.
Afghanistan dropped as many as five catches.
These included Young being dropped by Rahmat Shah off Fazalhaq Farooqi before he had scored in the first over while skipper Hashmatullah Shahidi let off Rachin Ravindra on nought and Latham on 38.
New Zealand were off to a slow start of 30 in the seventh over when Mujeeb Ur Rahman removed Devon Conway for 20.
Young, replacing skipper Kane Williamson who fractured his thumb in the last game, added 79 for the second wicket with Ravindra who scored a 41-ball 32.
But Afghanistan hit back with three wickets off just nine balls for one run as pacer Azmatullah Omarzai dismissed Ravindra and Young in the same over.
Young had three sixes and four boundaries in his knock.
In the next over spinner Rashid Khan had Daryl Mitchell for one as New Zealand lost their way.
It was left to Phillips, who cracked four sixes and as many boundaries, to join stand-in skipper Latham in a solid fifth wicket stand of 144 to revive the innings.
Latham hit three boundaries and two sixes but he and Phillips fell for the addition of just one run, leaving Chapman to give late impetus.
Fast bowlers Naveen-ul-Haq and Omarzai took two wickets apiece.
Brief scores:
New Zealand 288-6 in 50 overs (G. Phillips 71, T. Latham 68, W. Young 54) v Afghanistan
