The current method for assessing pain in clinical practice is subjective and relies on self-reported scales. An objective and accurate method of pain assessment is needed for physicians to prescribe the proper medication dosage, which could reduce addiction to opioids. Hence, many works have used electrodermal activity (EDA) as a suitable signal for detecting pain. Previous studies have used machine learning and deep learning to detect pain responses, but none have used a sequence-to-sequence deep learning approach to continuously detect acute pain from EDA signals, as well as accurate detection of pain onset. In this study, we evaluated deep learning models including 1-dimensional convolutional neural networks (1D-CNN), long short-term memory networks (LSTM), and three hybrid CNN-LSTM architectures for continuous pain detection using phasic EDA features. We used a database consisting of 36 healthy volunteers who underwent pain stimuli induced by a thermal grill. We extracted the phasic component, phasic drivers, and time-frequency spectrum of the phasic EDA (TFS-phEDA), which was found to be the most discerning physiomarker. The best model was a parallel hybrid architecture of a temporal convolutional neural network and a stacked bi-directional and uni-directional LSTM, which obtained a F1-score of 77.8% and was able to correctly detect pain in 15-second signals. The model was evaluated using 37 independent subjects from the BioVid Heat Pain Database and outperformed other approaches in recognizing higher pain levels compared to baseline with an accuracy of 91.5%. The results show the feasibility of continuous pain detection using deep learning and EDA.