Speight raced up the right wing to claim his second try of the match after Auckland had fought back from 12-0 down, with his score converted on the hooter.
The Brumbies started strongly when Tom Cox touched down within 35 seconds for the Brumbies with the extras added, before Speight scored their second five minutes later as the Australian side took a 12-0 lead at half-time.
Kali Hala then crossed under the posts for Auckland in the 11th minute after a rapid sprint up the pitch which was duly converted, before he added a second try a minute later following another strong run as the game looked to have been turned on its head.
Hala's try was converted from under the posts as Auckland took a 14-12 lead into the final two minutes.
However, it was Speight who had the final word, powering up the right touchline to dive in for the decisive try.
Earlier, Aviva Premiership's Harlequins took on Buenos Aires in the third-place play-off where the Argentinian side triumphed 31-7.
Buenos Aires took first blood through Ezequiel Valy with Santiago Alvarez scoring the second in the fifth minute before Lucas Alcacer added the third, with two conversions giving them a 19-0 lead at half-time.
Another score for Alcacer in the second half saw the Argentinians extend their lead further before Henry Cheeseman crossed for Quins, although his converted try was little more than a consolation as Gonzalo Gutuerrez Taboda scored to wrap up victory.
New York won the Plate final as they beat Gloucester 33-7.