failed to collect the ball, which instead fell at the feet of home defender Ricardo Costa who prodded it into the net in the 65th minute.
It was a triumph of last season's fifth-placed finishers over sixth and former Yugoslavia international Djukic joined Athletic Bilbao's Ernesto Valverde and Real Sociedad's Jagoba Arrasate in enjoying debut day wins.
Sociedad earlier swept aside Getafe 2-0 at home, and Bilbao were 2-1 victors at Real Valladolid.
La Liga champions Barcelona start out at home to Levante on Sunday when Real Madrid host Real Betis at the Bernabeu.
Valencia, who are struggling to come to terms with a crippling debt which forces them to sell their best players every summer, had the best of the chances.
Despite playing without last season's top scorer Roberto Soldado, who was sold to Tottenham Hotspur, they took the game to a Malaga side also struggling financially.
Qatar-owned Malaga have been banned from taking up their place in the Europa League this season by European soccer's governing body UEFA, for late payments to creditors.
Chances were limited with Malaga surprisingly resilient considering the changes from the team Manuel Pellegrini guided to the Champions League quarter-finals last term.
Once the visitors fell behind, however, they rarely looked like getting a goal back.
With the exodus of talent from a cash-strapped La Liga and worries the title race will be dominated more than ever by Real and Barca, the opening round of matches looked to be starting under a cloud.
However, Sociedad brightened the outlook with a pulsating win at Anoeta.
The San Sebastian-based side were surprise fourth-placed finishers in June and took the lead when Gonzalo Castro's first-time pass sent Mexico striker Carlos Vela through the middle to score with a dink over the keeper in the 42nd minute.
As Getafe chased the game, former Arsenal player Vela set new signing Haris Seferovic scampering clear down the left and the Switzerland striker also chipped the ball over Miguel Angel Moya to make it 2-0 from a tight angle in the 70th minute.
Vela should have had his second in the closing minutes but he miscued his shot when one-on-one with Moya.
"It was a textbook start. A win, no goals conceded and the team played well. It was positive," Arrasate told the club's website (www.valenciacf.com).
"It was important to start the league well and it gives us more confidence for the game on Tuesday," he added referring to their Champions League playoff first leg at Olympique Lyon.
Bilbao's win boasted two more cracking goals, both coming in three first-half minutes, as Markel Susaeta's opener, after a flowing passing move from the visitors, was cancelled out by Patrick Ebert's fierce equaliser for Valladolid.
Iker Muniain grabbed the winner five minutes after the break with Spain and Barcelona defender Gerard Pique prompted to tweet: "Great game in the Nuevo Zorrilla! Who said La Liga lacked quality or emotion?."