Ivan Toney coolly slotted in a 24th-minute penalty, awarded after Andros Townsend's high challenge on Frank Onyeka was reviewed by referee Darren England on the pitch-side monitor.
The goal ignited a largely uneventful start to a meeting between the two clubs rooted to the bottom of the Premier League's form table.
Salomon Rondon was denied an immediate equaliser by Bees goalkeeper Alvaro Fernandez at close range and Abdoulaye Doucoure produced a superb tackle to thwart Vitaly Janelt at the other end moments later.
Everton dominated following half-time but Rondon - replacing the suspended Richarlison in attack - saw an attempt blocked and Iwobi shot straight at Fernandez as the visitors chased an equaliser to no avail.
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