A large community of research has been developed in recent years to analyze social media and social networks, with the aim of understanding, discovering insights, and exploiting the available information. The focus has shifted from conventional polarity classification to contemporary application-oriented fine-grained aspects such as, emotions, sarcasm, stance, rumor, and hate speech detection in the user-generated content. Detecting a sarcastic tone in natural language hinders the performance of sentiment analysis tasks. The majority of the studies on automatic sarcasm detection emphasize on the use of lexical, syntactic, or pragmatic features that are often unequivocally expressed through figurative literary devices such as words, emoticons, and exclamation marks. In this paper, we propose a deep learning model called sAtt-BLSTM convNet that is based on the hybrid of soft attention-based bidirectional long short-term memory (sAtt-BLSTM) and convolution neural network (convNet) applying global vectors for word representation (GLoVe) for building semantic word embeddings. In addition to the feature maps generated by the sAtt-BLSTM, punctuation-based auxiliary features are also merged into the convNet. The robustness of the proposed model is investigated using balanced (tweets from benchmark SemEval 2015 Task 11) and unbalanced (approximately 40000 random tweets using the Sarcasm Detector tool with 15000 sarcastic and 25000 non-sarcastic messages) datasets. An experimental study using the training- and test-set accuracy metrics is performed to compare the proposed deep neural model with convNet, LSTM, and bidirectional LSTM with/without attention and it is observed that the novel sAtt-BLSTM convNet model outperforms others with a superior sarcasm-classification accuracy of 97.87% for the Twitter dataset and 93.71% for the random-tweet dataset.