Electroencephalogram (EEG) signals are crucial to Brain-Computer Interfacing (BCI). However, these are vulnerable to a variety of unintended artifacts that could negatively impact the precise brain function assessment. This paper provides a new algorithm to eliminate Electrical Shift and Linear Trend artifact (ESLT) in EEG using Singular Spectrum Analysis (SSA) and Enhanced local Polynomial (LP) Approximation-based Total Variation (EPATV). The contaminated single channel EEG is subdivided into multiple bands of frequency components by SSA. In order to acquire all LP and TV components, EPATV filtering is applied over the contaminated component frequency band. Filtered sub-signal is collected by subtracting both the LP and TV components from the component contaminated frequency band. Then, the addition of filtered sub-signal and remaining SSA frequency band components yield the final denoised EEG signal. The effectiveness of the proposed method in this paper is evaluated using the data obtained from three databases and compared with the existing methods. From the extensive simulation results, it is inferred that the algorithm discussed in the paper is effective when compared the existing methods, exhibiting a highest averaged Correlation Coefficient (CC) of 0.9534, averaged Signal to Noise Ratio (SNR) of 10.2208dB, lowest averaged Relative Root Mean Square Error (RRMSE) value 0.2787 and averaged Mean absolute Error (MAE) in α band value of 0.0557. The algorithm presented in this paper may be a viable choice for extracting ESLT artifact from a small streaming section of the EEG without requirement of the initial calibration or enormous EEG data.