The Croatia forward struck twice in Atletico's King's Cup game at home to third-tier Hospitalet on Thursday, a 2-2 second-leg draw that sent the 2013 winners through to the last 16 5-2 on aggregate and earned them a meeting with holders Real Madrid.
Atletico bought Mandzukic from Bayern Munich to replace Costa following the Brazil-born forward's departure for Chelsea in the close season.
While he may lack Costa's pace and ability to run at defenders, Mandzukic is equally strong in the air and his positioning and eye for goal make him lethal inside the penalty area.
He started his career in Spain with a bang when he scored the goal that won the Spanish Super Cup against Real in August but it took time for him to settle as Atletico sought to rebuild after the exits of Costa, fullback Filipe Luis and goalkeeper Thibaut Courtois.
In 13 La Liga appearances this term, Mandzukic has netted six goals, while in the Champions League his return is five in six games, helping Atletico top their group ahead of Juventus.
He has become a firm favourite with the Atletico fans and earned praise from Simeone for his double on Thursday.
"His enthusiasm and his ability have gelled with a team that plays in the same way," Simeone told a news conference.
"At the start maybe he found it a bit tough but he has become more involved with each match and his relationship with his team mates is improving all the time," added the Argentine, who has a reputation for getting the best out of his players.
"He has scored important goals, like the one in the Super Cup against Real Madrid, and his statistics are very positive for a forward.
"We hope he will continue to respond in the same way, and even get better."
Atletico's final game before the two-week winter break is Sunday's La Liga clash at Athletic Bilbao, when they will be looking to bounce back from last weekend's 1-0 reverse at home to Villarreal.
With 15 matches played, the champions are third on 32 points, three behind Barcelona and seven adrift of leaders Real.
(Editing by John O'Brien)
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