After asking Afghanistan to bat first, Neesham racked up his first maiden five-wicket haul as he returned career-best figures of 5/31 to bundle the opponents out for a paltry 172.
Chasing the modest target, Kiwis skipper Williamson led from the front with an unbeaten 79 off 99 balls, his innings laced with nine fours, as New Zealand won with 107 balls to spare.
Afghanistan lost star leg-spinner Rashid who was ruled out after their innings as a precautionary measure after the 20-year-old was hit on the head on the helmet as he got out in the first essay.
"Rashid Khan will take no further part in the game today. He has been pulled out as a precaution after being hit on his head while batting," the ICC said in a tweet.
Rashid was batting on 0 when a Lockie Ferguson delivery in the 34th over climbed into his helmet and ricocheted down onto the stumps.
Rashid was visibly shaken by the blow as Kane Williamson, his captain at Sunrisers Hyderabad in the Indian Premier League, was seen coming across and putting his arm around Rashid to check if he was alright.
The physio was then out on the field as Rashid was escorted off with his eyes starting to bulge.
Table toppers New Zealand have now won all three of their matches, having six points and a net run rate of +2.163.
Afghanistan, meanwhile, succumbed to their third straight defeat and remained at the foot of the 10-team points table.
The Afghans were off to a dream start when Aftab Alam (3/45) removed Martin Guptill in the very first delivery. Colin Munro (22) was also sent back by the pacer but once the experienced Williamson and Ross Taylor (48; 52b; 6x4; 1x6) joined hands for a 89-run stand for the third wicket, it was just a matter of time before the Black Caps sealed the issue.
Taylor was unfortunate to miss out on a half century as he got out to Alam but Tom Latham (13 not out) remained at the crease with Williamson who, fittingly, hit the winning runs as New Zealand won with 17.5 overs to spare putting up 173/3.
Earlier, Neesham broke Afghanistan's 66-run opening stand between Hazratullah Zazai and Noor Ali Zadran (31) by dismissing Zazai (34) before sending back Rahmat Shah (0) and then cleaning up their middle order with the scalps of skipper Gulbadin Naib (4), Mohammad Nabi (9) and Najibullah Zadran (4) in no time.
For the Afghans Hashmatullah Shahidi scored a defiant 59 but it was not enough to take them past the 200-run mark.
Besides Neesham, tearaway quick Ferguson took 4 wickets, giving away 37 runs in 9.1 overs. Ferguson ended Aftab Alam's 14 run cameo as he was caught by wicketkeeper Tom Latham who took as many as five catches behind the stumps.
Brief scores: Afghanistan: 172 all out (Hashmatullah Shahidi 59; James Neesham 5/31, Lockie Ferguson 4/37) vs New Zealand 173/3 in 32.1 overs (Kane Williamson 79 not out; Ross Taylor 48; Aftab Alam 3/45).