The Lions, under pressure to maintain their Super Rugby status after the South African Rugby Union guaranteed the Southern Kings promotion next year, were abject and only two tries in the final quarter narrowed the losing margin.
The Brumbies went behind early to a penalty by Lions flyhalf Elton Jantjies, but struck back with a 12th-minute try by prop Ruaidhri Murphy, who ploughed over the line from a ruck set up when Speight was stopped just short of the line after bursting down the blindside wing.
The visitors fell behind again, though, as Jantjies kicked another penalty four minutes later, and the Lions held on to the lead until the 22nd minute as Christian Lealiifano and fullback Jesse Mogg missed three kicks between them.
The Brumbies backs were a constant threat with their direct running and changes in the point of attack and Speight grabbed his first try after a ruck, from which Lealiifano made a half-break to present the wing with an easy touchdown.
Speight was involved again as the Brumbies scored five minutes before halftime, cutting in from his wing and breaking through the porous Lions defence and popping the ball to Andrew Smith to dot down.
The lacklustre Lions had defended poorly and kicked wastefully in the first half and started the second half in even worse fashion as they conceded two tries in the first seven minutes.
Lions fullback Andries Coetzee presented wing Joe Tomane with a welcome gift try as he failed to gather a Mogg grubber and flank Michael Hooper, having hassled the Lions all game at the breakdown, then punished them from the kickoff as he burst clear and set up prop Ben Alexander for a powerful finish.
Speight scored again for the Brumbies in the 58th minute after the Lions twice presented them with turnovers in the 22, stretching the lead to an unassailable 34-6.
The situation was made worse for the Lions in the 63rd minute when replacement centre Butch James charged into a ruck with his shoulder and was yellow-carded. But the home team finished the match as the stronger side, with centre Jaco Taute and loosehead prop Caylib Oosthuizen scoring late tries.
The Lions, coached by former All Black mentor John Mitchell, have now won only one of their eight matches and face a daunting task when they play the pace-setting Waikato Chiefs in Hamilton on May 5.
