Attentions now turn to South Africa for round two of the Sevens World Series, with less than a week's recovery time before they do it all again in Cape Town.
Tries from Andrew Knewstubb and Scott Curry brought New Zealand back into the match in the second period.
The Blitzboks made the ideal start to the defence of their HSBC World Sevens Series title by winning the season-opening tournament in Dubai on Saturday.
The reigning series champions fell to a 14-12 defeat after a late converted try handed the Americans a shock victory.
"We kind of let them back into the game", said Snyman.
"We deliberately didn't put the guys into anything like this in pre-season as it's such a long season for us so it was always going to be a little bit of a risk".
Rosko Specman, Seabelo Senatla and Kwagga Smith crossed to give the Blitzboks a 19-0 half-time lead before the All Blacks Sevens staged a valiant second-half rally.
"There were some pretty sticky performances from us which we knew would be the case with this being our first tournament of the season", said Amor.
Samuel Oliech stepped up to convert but Uganda offered an instant response through Solomon Okia who capitalized on poor handling from the Shujaa to score his fourth try in Dubai which was converted to leave the tie level at 7-7.
England picked up bronze after seeing off Fiji 28-21.
The men will close out their first day of competition against South Africa at 10:53 a.m. ET.
Kenya Sevens have qualified for today's Dubai Sevens Main Cup quarter-final after beating Uganda 29-14 in a physical decisive final Group A match at The Sevens Stadium, last evening. After an interception in their own half, New Zealand stormed forward through Sam Dickson but he got scragged short.