I get the appropriate values:

df = 3;

alpha = 0.01;
InverseCDF[ChiSquareDistribution[df], 1 - alpha]
(* 11.34486673014437` *)

alpha = 0.05;
InverseCDF[ChiSquareDistribution[df], 1 - alpha]
(* 7.814727903251181` *)

Also, there are many, many ways to mimic the calculator you reference. Here's one way:

Manipulate[
 InverseCDF[ChiSquareDistribution[df], 1 - \[Alpha]], 
 {{\[Alpha], 3}, {0.001, 0.005, 0.01, 0.05}},
 {{df, 1}, 1, 120, 1, Appearance -> "Labeled"}]

Does this work with any significance level or just 0.05? Got the same results when changed alpha to 0.01, probably doing something wrong. I am also trying to replicate this calculator in Mathematica, but keep getting an error with degree of freedom calculation.