"Dear Victoria, I congratulate you from all my soul on your remarkable victory in the Australian Open," Lukashenko said in a message to Azarenka, who left Belarus as a teenager to realise her talent and now lives in Monte Carlo.
"The whole country is proud of your achievement. Thanks sincerely to you for your brilliance and spirit and to your coaches for their tireless efforts to realise your talent," he said according to the official Belta news agency.
Azarenka, who was born in Minsk, had during the tournament talked up her Belarussian heritage and said she had taken inspiration from her grandmother, a former kindergarten teacher who worked into her 70s.
Belarus strongman Lukashenko is often pictured playing sports, notably ice hockey and cross country skiing, but is also condemned by rights activists for presiding over a ruthless crackdown on the opposition.
Azarenka earlier hammered the Russian former champion Maria Sharapova in an all ex-Soviet final 6-3, 6-0 in just 82 minutes, in her country's most significant sporting achievement for years.