The Spain international beautifully curled the ball into the top corner on the hour mark to take his tally to four goals in four La Liga games following his summer move from Villarreal.
Athletic, who have now lost their last three league games, once again enjoyed plenty of possession but lacked cutting edge up front and the hosts should have extended their lead but Eliseu shot into the side-netting after a wonderful run had presented him with an open goal.
Athletic coach Marcelo Bielsa has switched systems a number of times already this season in an attempt to find a winning formula and the Basque side, after returning to three at the back, enjoyed the bulk of possession in a first half lacking few clear-cut chances.
Eliseu had the best opportunity when a brilliant through-ball from Duda sent the Portuguese away with only Gorka Iraizoz to beat, but his lofted finish over the keeper cleared the goal.
Julio Baptista then struck the crossbar with a free-kick from the edge of the box that was to serve as a warning for Athletic before Markel Susaeta had the visitors' best chance of the half but delayed when well placed in the box to allow the Malaga defence to recover.
Malaga winger Joaquin Sanchez was first to threaten after the break for the hosts but he shot wide after cutting inside onto his left foot 10 minutes into the second half.
Athletic continually gave away free-kicks around their area and Cazorla finally made them pay with fierce strike that got up over the wall and dipped just in time to find Iraizoz's top corner.
Gaizka Toquero wasted Athletic's best chance of an equaliser when he dived to meet a cross from the left. Malaga goalkeeper Willy Caballero was able to easily push Toquero's effort over the bar but had the Athletic striker left the cross Fernando Llorente was standing in a much better position to make the telling finish.
Eliseu should have sealed all three points when he skipped past four Athletic challenges but gave himself too much to do by rounding Iraizoz and from a narrow angle could not hook the ball into the empty net.