Using analytical approach and Monte Carlo (MC) simulations, we study the elastic behavior of the intrinsically twisted elastic ribbons with bending anisotropy, such as double-stranded DNA (dsDNA), in two-dimensional (2D) confinement. We show that, due to the bending anisotropy, the persistence length of dsDNA in 2D conformations is always greater than three-dimensional (3D) conformations. This result is in consistence with the measured values for DNA persistence length in 2D and 3D in equal biological conditions. We also show that in two dimensions, an anisotropic, intrinsically twisted polymer exhibits an implicit twist-bend coupling, which leads to the transient curvature increasing with a half helical turn periodicity along the bent polymer.