This paper studies the existence and long time behavior of the solutions to the coupled Burgers-complex Ginzburg-Landau (Burgers-CGL) equations, which are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. This paper firstly shows the existence of the global solutions to these coupled equations via subtle transforms, delicate a priori estimates and a so-called continuity method, then prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor.