Australia also went through the mill in their semi-final and felt heartbreak at the death.
After several more minutes of intense attacking pressure, including two and a half more after the buzzer, New Zealand's incredible defence was rewarded with Scott Gregory exerting himself on the breakdown and winning a penalty.
Captain Sarah Goss scored three tries as New Zealand finished flawless in pool play on day one of the Dubai Sevens.
The Eagles opened the Sevens Series in Dubai by earning a spot in the title match against two-time defending World Cup champions New Zealand.
Scores were locked 21-all with time elapsed on the clock when Dick, playing in her first World Series tournament, gathered the ball 40-metres out from the line and powered away from the tiring United States of America defence for the decisive score on Friday.
Australian men's sevens coach Tim Walsh blamed a lack of composure at key moments for his team falling just short of a medal at the Dubai Sevens.
The teams went back and forth in the first half, New Zealand going to halftime 12-7 up after possession was ripped from Canada and Tyla Nathan-Wong scored.
The turning point came when Bianca Farella was sin-binned for a high tackle, and New Zealand took advantage with a scrappy run by Stacey Waaka and Nathan-Wong's second, insurance try.
A Harry Glover try with time running out lifted England to a 15-14 win over Australia in the third-place match. Sam Pecqueur scored a try with seconds left for Scotland to trail by only 21-19, then Jamie Farndale appeared to touch down the victor, but it was disallowed because he stepped out of bounds.
Connor Braid scored two tries and Luke McCloskey added a single in a losing cause against Australia.
That set up the final showdown with New Zealand, which advanced by blanking Russian Federation 31-0 and the usa 22-0 in the knockout rounds. They finished ninth overall on the circuit last season.
"We had two games where full time was all square and one we won (USA) and one we lost (Canada)".
It's a 10-even series, which includes the Canada Sevens from March 9-10 at B.C. Place. It was the Eagles' first time reaching the Dubai sevens final and thus their best finish in the prestigious season opener.
A top-four finish this season means automatic qualification for the 2020 Olympics.
Canada will face Zimbabwe in the consolation Challenge Trophy quarterfinal.