Hurricanes captain Dane Coles also crossed twice while James Blackwell scored after the final hooter for the home side, who were given no ball or space last week by the Crusaders, but were at their dynamic best as they played at pace and attacked from anywhere on the field.
Hooker Folau Fainga'a and Tom Wright crossed for the Brumbies, who showed signs of the play that destroyed the Waikato Chiefs 54-17 last week but were forced to chase the Hurricanes in the second half.
While both teams were eager to keep the ball alive, numerous turnovers meant neither were able to build much phase pressure.
The Hurricanes, who scored a barrage of points from turnover ball, opened a 14-0 lead after 15 minutes with Coles and Laumape both crossing after the home side capitalised on turnovers.
The hosts, however, were their own worst enemies conceding penalties on both attack and defence which not only relieved pressure on the Brumbies but also allowed the visitors to stay close on the scoreboard in the first half.
One such penalty led to an inevitable try from a driving maul, to Fainga'a, while Christian Lealiifano slotted a 29th-minute penalty after another Hurricanes' breakdown infringement.
That only served to reignite the Hurricanes and Coles added his second try after TJ Perenara produced a delightful no-look pass out of the back of his hand to give his side a 19-8 advantage at the break.
Beauden Barrett extended the lead as the Hurricanes dominated the third quarter with another Perenara no-look pass to Laumape - who smashed over two tacklers - leading to another try to give the hosts a 29-8 advantage.
The 25-year-old Laumape grabbed his third when Matt Proctor intercepted a Lealiifano pass and freed up his midfield partner to run 45 metres to score.
Wright crossed for a consolation try - his first in Super Rugby after switching from rugby league - in the 70th minute before another turnover after the hooter gave Hurricanes lock Blackwell his first career try.