Human telomeres as well as more than 40% of human genes near the promoter regions have been found to contain the sequence that may form a G-quadruplex structure. Other non-canonical DNA structures comprising bulges, hairpins, or bubbles may have a functionally important role during transcription, replication, or recombination. The guanine-rich regions of DNA are hotspots of oxidation that forms 7,8-dihydro-8-oxoguanine, thymine glycol, and abasic sites: the lesions that are handled by the base excision repair pathway. Nonetheless, the features of DNA repair processes in non-canonical DNA structures are still poorly understood. Therefore, in this work, a comparative analysis of the efficiency of the removal of a damaged nucleotide from various G-quadruplexes and bulged structures was performed using endonuclease VIII-like 1 (NEIL1), human 8-oxoguanine-DNA glycosylase (OGG1), endonuclease III (NTH1), and prokaryotic formamidopyrimidine-DNA glycosylase (Fpg), and endonuclease VIII (Nei). All the tested enzymes were able to cleave damage-containing bulged DNA structures, indicating their important role in the repair process when single-stranded DNA and intermediate non–B-form structures such as bubbles and bulges are formed. Nevertheless, our results suggest that the ability to cleave damaged quadruplexes is an intrinsic feature of members of the H2tH structural family, suggesting that these enzymes can participate in the modulation of processes controlled by the formation of quadruplex structures in genomic DNA.