The Michael addition of tetramethylsilyl azide to 3-methylenepiperidine-2,6-dione afforded new glutarimide derivative, 3-(azidomethyl)piperidine-2,6-dione, which was introduced into the CuAAC click reaction with a variety of alkynes to afford thirty novel structurally diverse 1,2,3-triazoles. The cytotoxicity of the synthesized compounds was evaluated on multiple myeloma cell lines (MM1.S, KMS-12-PE), a leukemia cell line (NALM-6), and normal B-cells (WIL2-S) showing a noticeable effect on the MM1.S cell line. Selected compounds demonstrated significant Cereblon binding affinity in a microscale thermophoresis assay with one derivative outperforming the reference drug Pomalidomide.