The 33-year-old's lead in the motorcycle class came during the 277-kilometre journey from Jeddah to Bisha in Saudi Arabia.
Price's placing was helped by a poor start from defending champion Ricky Brabec, who finished 24th and nearly 17 minutes behind his Australian rival.
The Red Bull Factory KTM rider admitted to making a navigational mistake during the opening stage, though it did little to prevent him from claiming his 13th career stage win at the Dakar.
"I got lost a little bit about seven kilometres from the finish. I took a little bit of time just trying to get back on track here," Price told the official race website.
"I think you're gonna see a lot of chopping and changing in the standings in this race. You just got to stay calm and let it cruise along and make it work."
Fellow Australian Daniel Sanders, meanwhile, dropped to 25th overall, trailing by almost 20 minutes.
SBS will broadcast extended daily highlights of the 2021 Dakar Rally - available from January 4-16 via SBS On Demand or watch on SBS nightly from 5:30 pm (AEDT).
