Honda was the most competitive bike in Stage 6, the last test of the rally before the mid-event rest day, as its riders topped the timesheets at every checkpoint.
Jose Ignacio Cornejo and Kevin Benavides were fastest early on but Brabec took over after 200 kilometres and never surrendered the lead afterwards.
Brabec's lead was never comfortable as Benavides was able to keep the gap at less than a minute, the duo separated by just 25s at the last checkpoint, 53km from the finish.
But then Benavides ground to a halt due to an engine problem and lost crucial time, allowing Brabec to top the stage.
Price, Brabec's main rival in the overall standings, had also hit trouble in the final part of the stage as he came to a standstill, costing the reigning champion about eight minutes.
Opening the road, Price was already far behind Brabec before his misfortune, but the incident caused him to end up 16m33s adrift.
The Australian fell to third overall, behind Pablo Quintanilla, in the general standings as the Husqvarna rider took fourth in the stage, only beaten by Brabec, Joan Barreda (Honda) and top KTM Matthias Walkner.
Price is also just two seconds ahead of Cornejo, as the Chilean and Barreda completed the top five at halfway point of the Dakar Rally.
Vitse dominates quads after Enrico stoppage
Simon Vitse dominated Stage 6 of the quad class but Ignacio Casale retained a dominant lead in the category as rival Giovanni Enrico was forced to retire.
While Casale fell out of victory contention early on, Vitse and Enrico were only half a minute apart in first and second in the first part of the stage.
The duel ended when Enrico's quad stopped with a mechanical problem at the 210km mark and he consequently retired.
Afterwards, Vitse was unrivalled and won the stage by 6m27s with Casale inheriting second.
Casale's lead is 35m12s at the halfway point of the rally and Vitse is the only rider within an hour of the Chilean. The top five is made up by Rafal Sonik, Alexandre Giroud and Manuel Andujar, the latter almost three hours adrift.
Romain Dutu, the fastest rider on the previous stage, crashed out early this time around.
Standings after Stage 6 (Bikes, top 10):
Jose Ignacio Cornejo