Linz collected a pass by Hoffer in the area and slid the ball past goalkeeper Andrei Sidelnikov to score one minute into stoppage time.
Hoffer, who came on as a substitute for Martin Harnik in the 66th, dribbled past Sidelnikov to double the lead in the next minute.
"I felt pretty bad after 88 minutes," Austria coach Dietmar Constantini said. "We needed a lot of patience. It made us insecure and we made a lot of mistakes."
Austria failed to convert several first-class scoring chances in the opening half hour.
Right winger Harnik caused danger with quick movements and several shots on goal, but most were just off-target.
Harnik presented Marc Janko with the team's best chance after 16 minutes but the captain shot well wide from inside the area.
Austria created hardly any opportunities afterward as play was slowed down by the defensive Kazakhs.
The only shot on target by Kazakhstan came in the 67th when goalkeeper Jurgen Macho saved a 20-meter attempt by Nurbol Zhumaskaliyev.
Austria and Kazakhstan, which lost its opening match against Turkey 3-0 four days ago, are in Group A with Germany, Belgium, Turkey and Azerbaijan.
Austria next plays Azerbaijan on Oct. 8 in Vienna.
Lineups:
Austria: Jurgen Macho, Franz Schiemer, Emanuel Pogatetz, Christian Fuchs, Jakob Jantscher (David Alaba, 66), Veli Kavlak, Sebastian Proedl, Ekrem Dag, Martin Harnik (Erwin Hoffer, 66), Roland Linz, Marc Janko (Stefan Maierhofer, 78).
Kazakhstan: Andrei Sidelnikov, Kazbek Geteriyev, Aleksandr Kirov, Renat Abdulin, Aleksandr Kislitsyn (Mikhail Rozhkov, 75), Andrei Karpovich, Evgeni Averchenko, Gleb Maltsev (Sergei Khizhnichenko, 46), Nurbol Zhumaskaliyev, Azat Nurgaliyev (Maksim Azovski, 59), Aleksei Popov.
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