The role of gas compression on the process of extremely fast flame acceleration in obstructed cylindrical tubes is studied analytically and validated by computational simulations. The acceleration leading to a deflagration-to-detonation transition is associated with a powerful jet-flow produced by delayed combustion in spaces between the obstacles. This acceleration mechanism is Reynolds-independent and conceptually laminar, with turbulence playing only a supplementary role. In this particular work, the incompressible formulation [Combust. Flame 157 (2010) 1012], Ref. 15 is extended to account for small but finite initial Mach number up to the first-order terms. While flames accelerate exponentially during the initial stage of propagation, when the compressibility is negligible, with continuous increase in the flame velocity with respect to the tube wall, the flame-generated compression waves subsequently moderate the acceleration process by affecting the flame shape and velocity, as well as the flow driven by the flame. It is demonstrated that the moderation effect is substantial, and as soon as gas compression is relatively small, the present theory is in good quantitative agreement with the computational simulations. The limitations of the incompressible theory are thereby underlined, and a critical blockage ratio for with this acceleration mechanism can be evaluated.