Another promoted club, Real Valladolid, also started with a victory, winning 1-0 at Real Zaragoza, while Rayo Vallecano beat Granada by the same scoreline thanks to a dramatic 95th-minute free-kick from Roberto Trashorras.
Depor's Riki cut in off the right flank and beat Osasuna goalkeeper Andres Fernandez at his near post in the 53rd minute, making up for a penalty miss just before the break when he fired against the post.
Jose Luis Oltra's side, who had made most of the running at a festive Estadio Riazor after a year in the second division, were penned back in their penalty area in time added on before Portugal forward Oliveira broke away to lob the ball over Fernandez for their second.
Along with Valencia, Depor are one of only two teams to have denied Real Madrid and Barcelona the Spanish league title in the last 16 years, and were regulars in European competition before sliding to relegation in 2011.
The win put them third in the standings, behind leaders Barca, who thrashed Real Sociedad 5-1 at home on Sunday, and Real Betis, who overcome Athletic Bilbao 5-3 in an eight-goal thriller at San Mames.
"We aren't going to get carried away with ourselves. The objective remains the same," Oltra said in reference to their stated aim of securing their top-flight future.
"The important thing is we have achieved a good result and started well. It is going to be a long and difficult year."
Valladolid were the brighter side at the Romareda and Oscar stabbed them in front from close range after helping set up the move just before the break.
Zaragoza goalkeeper Roberto kept the hosts in the game tipping a Victor Perez penalty on to the post in the 67th minute and joined his side in attack in the frantic finale, but to no avail.
It was a case of déjà vu in Vallecas as Rayo Vallecano snatched all the points against Granada when Trashorras struck a superb freekick into the top corner in added time.
Rayo won by the same 1-0 scoreline in front of their own fans on the final day of last season against Granada, when Raul Tamudo also netted in time added on to save them from relegation and send Villarreal down instead.