Juan Rodriguez and Pablo Amo took advantage of sloppy marking give Deportivo the win in the second half, leaving Barca third on 61 points, one behind Villarreal who visit Real Betis on Sunday, and 11 adrift of leaders Real Madrid.
If Villarreal fail to beat Betis and Real win at home to Athletic Bilbao in Sunday's late game, Bernd Schuster's side will retain their league crown with four matches left to play.
Depor's sixth win in seven put them seventh, three points behind Sevilla, who missed out on a chance to close in on the Champions League places with a 0-0 draw at relegation-threatened Real Murcia in the late game.
|"My decision is not influenced by being first or second or still in the Champions League."|
Rijkaard left Lionel Messi, Xavi and Samuel Eto'o out of the squad, and with Gabriel Milito and Andres Iniesta suspended, it was an unfamiliar line up who started in La Coruna.
"We put a team out there to do a job but before thinking about the result, I think about the health of my players," Rijkaard explained.
"My decision is not influenced by being first or second or still in the Champions League.
"Eto'o and Xavi are both in need of a rest to gain strength. As for Messi his case is different."
Messi not ready
Argentine winger Messi, 20, played in the first leg against Man United but has only just returned to first team action after a torn muscle in his left leg back in March interrupted his season.
"Some people are saying Messi is ready to play ninety minutes, but unfortunately that is not the case," Rijkaard added.
"He has already suffered two very serious injuries and we must protect him - I no longer speak as the coach of Barca, but as a fan of football, because I would like to see Messi playing for many years.
"I do not rest anyone thinking about another game, but only because the player needs it."
The visitors dominated possession without managing to open up the packed home defence in the first half, and it was Depor who had the first chance on goal when reserve goalkeeper Jose Manuel Pinto fumbled a cross.
In the scramble that followed, Xisco lashed a shot against the bar and Rodriguez sent a shot wide of the post with the goal wide open.
Bojan Krkic, teenage striker, provided Barca's greatest threat with a couple of efforts, but Rodriguez made amends for his earlier miss, slotting home Angel Lafita's pull back from the touchline just after the break.
The home side took control of the game and defender Amo was left unmarked at the back post to head in a free kick as Barca made it one win from nine in the league.