Atomistic modeling of the optoelectronic properties of organic semiconductors (OSCs) requires a large number of excited-state electronic-structure calculations, a computationally daunting task for many OSC applications. In this work, we advocate the use of deep learning to address this challenge and demonstrate that state-of-the-art deep neural networks (DNNs) are capable of accurately predicting various electronic properties of an important class of OSCs, i.e., oligothiophenes (OTs), including their HOMO and LUMO energies, excited-state energies and associated transition dipole moments. Among the tested DNNs, SchNet shows the best performance for OTs of different sizes, achieving average prediction errors in the range of 20-80 meV. We show that SchNet also consistently outperforms shallow feed-forward neural networks, especially in difficult cases with large molecules or limited training data. We further show that SchNet could predict the transition dipole moment accurately, a task previously known to be difficult for feed-forward neural networks, and we ascribe the relatively large errors in transition dipole prediction seen for some OT configurations to the charge-transfer character of their excited states. Finally, we demonstrate the effectiveness of SchNet by modeling the UV-vis absorption spectra of OTs in dichloromethane, and a good agreement is observed between the calculated and experimental spectra.