Chasing their first tournament win since Vancouver in 2016, the New Zealanders clawed their way back with tries to Scott Curry and Andrew Knewstubb after Specman was yellow-carded for a deliberate knock on.
Defending series' champions South Africa won the tournament beat the All Blacks Sevens 24-12 in the decider.
The season curtain-raiser saw England dusting off the cobwebs but they still did enough to win three matches from three in the HSBC World Rugby Sevens Series - setting up a last-eight tie against rivals Australia.
South Africa made a blistering start to the final, with Philip Snyman, Senatla and Kwagga Smith scoring tries in the opening four minutes of the contest.
"It was a great performance from the guys", said Senatla.
Besides Senatla, who starred throughout the two days in Dubai, Werner Kok also stood out in a team that again looked well-coached, fit and happy.
The next round is in South Africa next weekend.
"It's easy to play well in a team as good as this‚" Smith said.
Amor reserved particular praise for Harry Glover who scored the last second winning try against Australia.
Eden Agero put restored Kenya's lead thanks to some quick thinking as he rose off the ruck unnoticed after tapping against the ball and then racing clear of two Ugandan clutches score to make it 12-7.
They had earlier finished pool play undefeated, accounting for South Africa 40-0 and France 28-17 alongside their comprehensive win over the US. After an interception in their own half, New Zealand stormed forward through Sam Dickson but he got scragged short.