Jose Gimenez's first-half strike sent the visitors on their way to victory and Mandzukic's eighth goal in nine matches across all competitions secured all three points just after the break.
Atletico did find their hosts hard to break down on occasions at Estadio Martinez Valero on Saturday, although it came as no surprise when their opening goal arrived after a well-worked set-piece.
Fran Escriba's men appeared more fluid going forward right at the start of the second period, but that only seemed to play into Atletico's hands, as Mandzukic finished a swift counterattack by claiming his sixth La Liga goal.
The result means Atletico's formidable defensive partnership of Uruguayan duo Gimenez and Diego Godin claimed the club's fourth consecutive clean sheet in all competitions.
Atletico cleverly worked space for themselves down the left following a short corner and, after receiving Arda Turan's cut-back, Gimenez lashed a fierce half-volley past Przemyslaw Tyton in emphatic fashion.
Diego Simeone's men tightened their grip on proceedings after taking the lead, with Gabi and Tiago dictating the play in the heart of the Atletico midfield.
However, Elche did go close to drawing level unexpectedly in the 34th minute, as Faycal Fajr's deep cross appeared to be heading for the top-left corner, only for Miguel Angel Moya to claw it out from underneath the crossbar.
Gabi dropped an excellent lofted pass over the hosts' defence for Mandzukic and the Croatian coolly slotted past Tyton after holding off David Lomban eight minutes into the second period.
Elche did force Moya into action with 16 minutes to go as the goalkeeper kept out Cristian Herrera's deflected effort, but Atletico had little trouble in absorbing the hosts' late pressure and the victory keeps the pressure up on leaders Real Madrid.
