Out of various moving average filter (MAF)-based phase-locked-loop (PLL), quasi type-1 PLL (QT1-PLL) is widely adopted due to its fast dynamic performance, implementation simplicity, and harmonics rejection abilities. However, the performance of QT1-PLL deteriorates in the presence of an off-nominal frequency unbalanced grid voltage component. Moreover, the sensitivity towards the fundamental frequency negative sequence (FFNS) component is high. Hence, this paper proposes a novel enhanced QT1-PLL solution that is insensitive to unbalance in the grid voltage signal during off-nominal frequency conditions. The proposed adaptive phase detector makes it possible to estimate both the fundamental frequency positive sequence (FFPS) and FFNS components with a high degree of immunity against harmonics. Notably, the pre-loop separation of the FFPS and the FFNS components helps suppress the second harmonic oscillations for improving the parameter estimation accuracy. The loop-filter design of QT1-PLL remains unaffected and requires a proportional gain to estimate the fundamental phase and frequency information. To address the DC offset issue, a modified delayed signal cancellation method is also proposed, which can theoretically eliminates the DC offset for any arbitrary delay length. A small-signal model of the proposed PLL is developed for the sake of stability analysis. Comparative results are provided