Laser wakefield acceleration is a frequently utilised research methodology for enhancing the energy levels of lighter charged particles, specifically electrons, to relativistic magnitudes. In this investigation, we utilised a linear polarised Gaussian-like laser pulse that propagated along the z-axis through cold collisionless underdense plasma in weakly nonlinear regime. An external planer magnetic wiggler field is applied along the y-axis. The influence of various critical parameters, such as amplitude and propagation constant of wiggler magnetic field, amplitude of laser electric field and laser pulse length on the wakefield and electron energy gain has been studied. A wiggler-assisted laser wakefield accelerator, the electron energy and wakefield evolution can be tuned by the wiggler magnetic field strength. The numerical findings demonstrate that by varying the strength of wiggler magnetic field and laser electric field, the amplitude of the wakefield is affected significantly. Furthermore, the equality of the order of pulse length and plasma wavelength is essential to obtain energy efficient acceleration mechanism. By employing specific parameters, a maximum energy increase of 2.26 GeV is achieved. This research will aid in the development of an energy-efficient electron acceleration technology by choosing suitable laser and plasma parameters.