Astana became the first Kazakh side to reach the knockout stages of a European competition and face Sporting Lisbon in the Europa League round of 32 next month.
"We have achieved a lot together over the years... and I hope that we will be able to stand up for the honour of our club in matches against Sporting," Stoilov told Astana's website.
Stoilov, who has twice coached the Bulgarian national team, also managed Levski Sofia in 2006 when they became the first team from the Balkan country to compete in the Champions League group stages.
(Reporting by Angel Krasimirov, editing by Ed Osmond)
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