Scott Rogers’ charges were full value for their victory at Huish Park and an Alex Greenwood penalty and a goal from captain Gemma Bonner completed the scoring.
The Reds made a bright start to proceedings and opened the scoring on 23 minutes, when Shanice van de Sanden’s shot was handled in the penalty area and Greenwood put away the spot-kick.
The second goal arrived from a corner just before the break, Greenwood turning provider as her delivery was turned home by Bonner.
Zelem was then introduced from the bench and she soon made it 3-0 on 58 minutes with a great finish from Van de Sanden’s cross.
Rogers handed new signing Jess Clarke a debut, before another Van de Sanden assist led to Zelem making it four.
Yeovil netted a late consolation through a Sarah Wiltshire penalty, but that took nothing away from an impressive Liverpool display.
Liverpool Ladies: Chamberlain, Hodson, Greenwood, Ingle, Bonner, Longhurst (Rodgers, 81), Weir (Zelem, 58), Coombs, Charles (Clarke, 71), Harding, Van de Sanden.