The Dutchman snatched the winner after a lacklustre performance by the hosts, who last season failed to win their first league title in 12 years after slipping up on the last matchday.
Dortmund were in control but hardly managed to get into the Cologne box.
The visitors had their best chance in the first half with Davie Selke's deflected shot that bounced off the cross bar.
There was little to cheer for the sold-out 81 000 crowd in the stadium, with Dortmund finding no way past a disciplined Cologne defence for much of the game, even after coach Edin Terzic replaced his entire frontline in the second half.
It was Cologne who came close in the second half with substitute Sargis Adamyan's header blocked by keeper Gregor Kobel as the Dortmund fans started jeering and whistling.
But Malen broke the deadlock, volleying in at the far post after Felix Nmecha had headed on a corner, to give his team a winning start to the campaign.
Champions Bayern Munich kicked off their season with a 4-0 demolition of hosts Werder Bremen on Friday with league record signing Harry Kane scoring one goal and setting up another on his Bundesliga debut.