Yilmaz's deflected shot put Turkey ahead in the 15th minute and the 35-year-old Lille forward struck his second in the 34th minute before sealing his hat-trick with an 81st-minute penalty.
In the opening game of their campaign to defend their World Cup title, France were held to a 1-1 draw by Ukraine in Paris.
Antoine Griezmann's opener was cancelled out by a Presnel Kimpembe own goal in the 57th minute, and Les Blues were unable to find a winner.
West Ham's Tomas Soucek struck a hat-trick as the Czech Republic stormed to a 6-2 win over Estonia.
Estonia had taken an early lead through Rauno Sappinen, but the Czechs hit back six minutes later, before going ahead shortly afterwards with goals from Patrik Schick and Antonin Barak.
Soucek then scored three times between the 32nd and 48th minutes to give the Czech Republic a four-goal cushion. Jakub Jankto scored his side's sixth goal before Estonia reduced the deficit when substitute Henri Anier scored four minutes from time.
Harry Wilson stunned the Red Devils with a superbly-crafted 10th minute goal, but the star-studded hosts were soon ahead as De Bruyne fired home from distance and then played a major part as Thorgan Hazard's header gave them the lead. Romelu Lukaku's penalty sealed victory for Roberto Martinez's side.
Mitrovic struck twice, superbly chipping the goalkeeper before heading in a second to become his country's joint-top scorer.
Alan Browne had given Ireland an 18th-minute lead, only for Dusan Vlahovic to equalise. Mitrovic's double followed, with James Collins' late strike mere consolation for the visitors.
Elsewhere, Slovenia secured a surprise 1-0 win over 2018 finalists Croatia.
Finland drew 2-2 with Bosnia and Herzegovina, while Portugal beat Azerbaijan 1-0.
Norway secured a comfortable 3-0 win over Gibraltar, while Russia were 3-1 winners over Malta.
Montenegro won 2-1 against Latvia and Cyprus and Slovakia played out a goalless draw.