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We present a $D$-dimensional Bardeen like Anti-de Sitter (AdS) black hole solution in Einstein-Gauss-Bonnet (EGB) gravity, \textit{viz}., Bardeen-EGB-AdS black holes. The Bardeen-EGB-AdS black hole has an additional parameter due to charge ($e$), apart from mass ($M$) and Gauss-Bonnet parameter ($\alpha$). Interestingly, for each value of $\alpha$, there exist a critical $e = e_E$ which corresponds to an extremal regular black hole with degenerate horizons, while for $e< e_E$, it describes non-extremal black hole with two horizons. Despite the complicated solution, the thermodynamical quantities, like temperature ($T$), specific heat($C$) and entropy ($S$) associated with the black hole are obtained exactly. It turns out that the heat capacity diverges at critical horizon radius $r_+ = r_C$, where the temperature attains maximum value and the Hawking-Page transition is achievable. Thus, we have an exact $D$-dimensional regular black holes, when evaporates lead to a thermodynamical stable remnant.