<p>In this paper, we study a two-phase flow system consisting of the compressible pressureless Euler equations and the incompressible Navier-Stokes equations coupled through the drag force. Owing to the lack of a pressure term in the Euler equations, the classical existence theory for hyperbolic-parabolic systems cannot be applied to this particular system. To solve this problem, we employ the spectral analysis method. At the beginning, we formulate the energy estimates for the velocities. Subsequently, in order to acquire the uniform estimates of the density, we make use of the spectral analysis method to derive a more rapid time-decay rate for the velocity of the Euler flow, specifically <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\|\nabla u\|_{H^3}\leq C(1+t)^{-\frac{5}{4}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msub> <mo>∥</mo> <mrow> <msup> <mi>H</mi> <mn>3</mn> </msup> </mrow> </msub> <mo>≤</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation>. After obtaining the above decay estimate, we manage to prove the uniform boundedness of the density. Eventually, leveraging the uniform estimates of the velocities and the continuity argument, we establish the global well-posedness of this two-phase flow system. Meanwhile, it is also proved that the velocities (<i>u, v</i>) decay to the motionless state at the optimal algebraic time-decay rates in <i>L</i>²-norm. Our results suggest that the smoothing effect of Navier-Stokes equations can be propagated to Euler equations through the drag force.</p>