A try from replacement lock Jordan Smiler gave the home side an 8-6 edge at the break but it took two second half interventions from Speight to break open an arm-wrestle of a contest dominated by defence.
A 67th minute try from loose forward Lappies Labuschagne off the back of a trademark Bulls rolling maul gave the visitors hope of a comeback victory but the Brumbies held firm in a tense 13 minutes.
The victory lifted the Brumbies to the top of the Australian conference ahead of reigning Super Rugby champions New South Wales Waratahs, who play the Lions in Johannesburg on Saturday.
Flyhalf Handre Pollard kicked three penalties and converted the try but it was not enough to prevent the Bulls from falling to their third straight defeat on their Australasian road trip.
The twice champions will probably have to break a 10-match losing streak outside South Africa going back to March 2013 in their penultimate match of the season against the Melbourne Rebels next week to stay in the playoff hunt.
Pollard's second penalty had given the South Africans a 6-0 lead when Smiler came on in the 24th minute for Sam Carter after the Wallabies lock suffered what looked like a serious knee injury.
Six minutes later the Brumbies finally got some sustained forward momentum and scrumhalf Nic White's quick ball from a ruck gave Smiler the room to get over the try line.
Pollard's third penalty got the visitors back in front straight after the break but four minutes later Speight took the ball at first receiver from an attacking scrum and showed his pace and power to burst through the Bulls midfield and score.
Speight's second try came 12 minutes later when the Brumbies got the ball out wide and the winger shrugged off the tackle of JJ Engelbrecht to touch down.
(Reporting by Nick Mulvenney, editing by Sudipto Ganguly)
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