Controllable fracture propagation in glass can be obtained by locally weakening the material. The high aspect ratio laser-induced modifications, required for thick glass dicing, can be obtained by use of Bessel-like beams, which have a long non-diffractive length with a small quasi-propagation-invariant central core. We demonstrate the possibility to form elongated modifications in glass with dominant transverse crack propagation direction using axicon-generated asymmetrical Bessel beam. Furthermore, the axicon tilt operation is introduced to add additional astigmatic aberrations for Bessel beam asymmetry control. Dicing results were compared using symmetrical and asymmetrical axicon-generated Bessel beams.