The paradigm for gamma-ray burst (GRB) prompt emission is changing. Since early in the Compton Gamma Ray Observatory (CGRO) era, the empirical Band function has been considered a good description of the keV-MeV γ-ray prompt emission spectra despite the fact that its shape was very often inconsistent with the theoretical predictions, especially those expected in pure synchrotron emission scenarios. We have recently established a new observational model analyzing data of the NASA Fermi Gamma-ray Space Telescope. In this model, GRB prompt emission would be a combination of three main emission components: (i) a thermal-like component that we have interpreted so far as emission from the jet photosphere, (ii) a non-thermal component that we have interpreted so far as either synchrotron radiation from the propagating and accelerated charged particles within the jet or reprocessed jet photospheric emission, and (iii) an additional non-thermal (cutoff) power law (PL) extending from low to high energies in γ-rays and most likely of inverse Compton origin. In this article we reanalyze some of the bright GRBs, namely GRBs 941017, 970111, and 990123, observed with the Burst And Transient Source Experiment (BATSE) on board CGRO with the new model. We conclude that BATSE data for these three GRBs are fully consistent with the recent results obtained with Fermi: some bright BATSE GRBs exhibit three separate components during the prompt phase with similar spectral parameters as those reported from Fermi data. In addition, the analysis of the BATSE GRBs with the new prompt emission model results in a relation between the time-resolved energy flux of the non-thermal component, F i nTh , and its corresponding νFn spectral peak energy, E peak,i nTh (i.e., F i nTh -E peak,i nTh ), which has a similar index-when fitted to a PL-as the one initially derived from Fermi data. For GRBs with known redshifts (z) this results in a possible universal relation between the luminosity of the non-thermal component, L i nTh , and its corresponding νFn spectral peak energy in the rest frame, E peak,i NT,rest (i.e., L i nTh -E peak,i NT,rest ). We estimated the redshifts of GRBs 941017 and 970111 using GRB 990123-with z = 1.61-as a reference. The estimated redshift for GRB 941017 is typical for long GRBs and the estimated redshift for GRB 970111 is right in the range of the expected values for this burst.