Organic solar cells (OSCs) based on a small molecule donor (S_D) and a polymer acceptor (P_A) exhibit low power conversion efficiency (PCE) due to the limited number of small molecule donor–polymer acceptor combinations. In this work, we employ a polymer acceptor based on the double B ← N bridged bipyridyl (BNBP) unit to develop S_D/P_A-type OSCs. With poly[(N,N′-bis(2-hexyldecyl)-diamine-bis(difluoro-borane)-2,2-bipyridine)-alt-(2,5-thiophene)] (P-BNBP-T) as the acceptor and 7,7′-(4,4-bis(2-ethylhexyl)-4H-silolo[3,2-b:4,5-b′]dithiophene-2,6-diyl)bis(6-fluoro-4-(5′-hexyl-[2,2′-bithiophen]-5-yl)benzo[c][1,2,5]thiadiazole) (p-DTS(FBTTh₂)₂) as the donor, the OSC device shows a high open-circuit voltage (V_OC) of 1.08 V and a PCE of 3.50%. The V_OC is ca. 0.3 V greater than that of other OSCs based on p-DTS(FBTTh₂)₂ due to the larger offset between the HOMO energy level of p-DTS(FBTTh₂)₂ and the higher-lying LUMO energy level of P-BNBP-T. The PCE of p-DTS(FBTTh₂)₂/P-BNBP-T is higher than that of any other OSCs based on the p-DTS(FBTTh₂)₂/polymer acceptor blend reported so far. These results indicate that the BNBP-based polymer acceptors are promising for high-performance S_D/P_A-type OSCs. While the as-cast p-DTS(FBTTh₂)₂/P-BNBP-T blend film exhibits low molecular packing order and large-size phase separation, processing with solvent additive 1,8-diiodoctane (DIO) leads to continuous networks with small crystalline grains of p-DTS(FBTTh₂)₂ in the blend film. The resulting OSC device exhibits the best photovoltaic performance because of the improved exciton dissociation efficiency and charge transport ability.