The 319km opening stage of the 2020 Dakar Rally was run between Jeddah and Al Wajh.
Brabec, who suffered a bike failure from the lead last year, led for about the first 100 kilometres of the stage before Price took over.
The American kept the gap close for the whole stage, with Price only able to record the stage win by 2m05s.
Both of Price's star teammates Matthias Walkner and Sam Sunderland, champions in 2018 and 2017 respectively, were close to the top throughout the day, with the former ending up third, 2m40s adrift.
Sunderland was further a 2m35s behind and is only fifth in the order, behind Honda's Kevin Benavides.
The Briton is closely ahead of top Husqvarna man Pablo Quintanilla, followed by the third Honda of Joan Barreda.
Yamaha had a low-key start to the rally as French duo Adrien van Beveren and Xavier de Soultrait slotted in 10th and 13th respectively.
Casale dominates quad class
Ignacio Casale controlled the quad class on the Chilean's return to the category, as Axel Dutrie, his closest rival initially, crashed out.
Casale, champion in 2014 and 2018, switched to UTV's last year but returned to quads for 2020, and has led the entire opening stage of the event.
Dutrie was closest to him early on but even his deficit was over five minutes already before they were 100 kilometres into the stage.
But Dutrie then crashed out at the 134km mark, the Frenchman treated by the medical team as he suffered his third consecutive retirement from the Dakar Rally.
Afterwards, 2015 champion Rafal Sonik stepped up as Casale's main challenger but the Chilean kept pulling away, topping the stage by 5m36s.
The top five was completed by Tomas Kubiena, Giovanni Enrico and Manuel Andujar.
Standings after Stage 1 (Bikes, top 10):
|10||Adrien van Beveren||Yamaha||9m52s|