The Frenchman’s strikes in the ninth and 27th minutes sent West Brom on their way, with half-time sub Graham Dorrans scoring the third from the spot nine minutes from time.
Anelka, who left Chelsea in January 2012 for Chinese side Shanghai Shenhua, was signed by the Baggies earlier this month – his sixth Premier League club – and started up front with Shane Long, who missed Thursday’s 2-1 defeat by Hannover with a toe injury.
He poked home the first after the Hungarians failed to clear a Chris Brunt free-kick while the second was easier on the eye, the Frenchman heading home a Markus Rosenberg cross having stylishly fed the Swede.
Kemar Roofe was one of several further players introduced on the hour mark by Steve Clarke – with Anelka leaving the fray – and he won the penalty which Dorrans tucked away.
The win concludes Albion's pre-season trip, with games against Derby, Atromitos, Genoa, Cork City and Bologna yet to come for the first team back in England.