The 63-year-old Dutchman, who led Russia to the semi-finals of Euro 2008 during his four years in charge, will leave when his contract expires on June 30 and take charge of Turkey in August.
"I'm really glad and happy that my last game in charge as Russia coach was against a friend of mine and a fellow Dutchman (Erwin Koeman)," Hiddink told reporters.
Koeman's Hungary took the lead in the 39th minute against the run of play.
Russia defended a corner poorly and left back Vilmos Vanczak squeezed a shot through the hands of keeper Igor Akinfeyev to grab his first international goal on his 45th appearance.
Russia equalised in the 59th minute when a corner from substitute Andrei Arshavin bounced into the net off Diniyar Bilyaletdinov's knee.
The hosts pushed for a winner in the last 10 minutes but Adam Szalai's powerful header and Laszlo Bognar's effort were well saved by substitute keeper Vladimir Gabulov.
Russian Igor Denisov was sent off in the last minute.
Hungary have not beaten Russia since their 2-0 win in Budapest in 1978.
Neither Russia nor Hungary have qualified for the World Cup in South Africa in June.
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