Despite the proliferation of numerous deep learning methods proposed for generic ECG classification and arrhythmia detection, compact systems with the real-time ability and high accuracy for classifying patient-specific ECG are still few. Particularly, the scarcity of patient-specific data poses an ultimate challenge to any classifier. Recently, compact 1D Convolutional Neural Networks (CNNs) have achieved the state-of-the-art performance level for the accurate classification of ventricular and supraventricular ectopic beats. However, several studies have demonstrated the fact that the learning performance of the conventional CNNs is limited because they are homogenous networks with a basic (linear) neuron model. In order to address this deficiency and further boost the patient-specific ECG classification performance, in this study, we propose 1D Self-organized Operational Neural Networks (1D Self-ONNs). Due to its self-organization capability, Self-ONNs have the utmost advantage and superiority over conventional ONNs where the prior operator search within the operator set library to find the best possible set of operators is entirely avoided. As the first study where 1D Self-ONNs are ever proposed for a classification task, our results over the MIT-BIH arrhythmia benchmark database demonstrate that 1D Self-ONNs can surpass 1D CNNs with a significant margin while having a similar computational complexity. Under AAMI recommendations and with minimal common training data used, over the entire MIT-BIH dataset 1D Self-ONNs have achieved 98% and 99.04% average accuracies, 76.6% and 93.7% average F1 scores on supra-ventricular and ventricular ectopic beat (VEB) classifications, respectively, which is the highest performance level ever reported.