We revisit charge density wave (CDW) behavior in the archetypal quasi-one-dimensional (quasi-1D) material NbSe3 by high-resolution angle-resolved photoemission spectroscopy measurements utilizing a microfocused laser with a photon energy of 6.3 eV. We present a detailed view of the electronic structure of this complex multiband system and unambiguously resolve CDW gaps at the Fermi level (EF). By employing a tight-binding model, we argue that these gaps are the result of interband coupling between electronic states that reside predominantly on distinct 1D chains within the material. Two such localized states are found to couple to an electronic state that extends across multiple 1D chains, highlighting the importance of a higher-dimensional interaction in stabilizing the CDW ordering in this material. In addition, the temperature evolution of intrachain gaps caused by the CDW periodicities far below EF deviate from the behavior expected for a Peierls-type mechanism driven by nesting; the upper and lower bands of the renormalized CDW dispersions maintain a fixed peak-to-peak distance while the gaps are gradually removed at higher temperatures. This points toward a gradual loss of long-range phase coherence as the dominant effect in reducing the CDW order parameter, which may correspond to the loss of coherence between the coupled chains. Furthermore, one of the gaps is observed above the bulk and surface CDW transition temperatures, implying the persistence of short-range incoherent CDW order. The influence of such higher-dimensional interactions likely plays an important role in a range of low-dimensional systems.