The Lazio striker opened the scoring during the 2014 World Cup qualifying win in Munich, taking his international tally to 68 goals from 129 appearances - level with striking legend Muller.
Toni Kroos and Thomas Muller added a goal each to consolidate Germany's position at the top of Group C but the day belonged to Klose, who remained adamant that it was farcical to compare him to the Bayern Munich and Mannschaft idol.
The 35-year-old told reporters after the victory: "It means an awful lot, but I do not want you to put me on the same level as Gerd. It's an absolute joke to compare myself with him.
"One must not compare him with anyone after what he has done. The number of matches and goals, he is unique."
Klose's coach, Joachim Low, believes extra preparation time gave his Germany team an immediate advantage.
He continued: "We had two or three days more time this week [than usual] and did our homework. We knew how Austria play and have made them toothless because our defence played well. Ultimately, it's the victory that counts."