Myrie-Williams netted after an error by goalkeeper Chris Neal had handed the visitors the lead at Vale Park in League Two.
Much of the first-half was a hard fought and scrappy affair, with Neal and visiting goalkeeper Chris Lewington largely redundant.
Vale did build up a head of steam towards half-time as Ashley Vincent and then two attempts from Tom Pope finally warmed Lewington's fingers.
But it was the visitors who took the lead following a 72nd-minute mistake by Neal, who was Vale's hero in Saturday's victory at Bradford. Luke Howell's powerfully struck 30-yard shot was fired straight at Neal, but the ball slipped through his fingers and rolled over the line.
Howell then blasted a much easier chance over the top from 12 yards just seconds later and the miss proved costly as Myrie-Williams levelled in the 75th minute.
Dagenham failed to clear from a goal-mouth scramble and the Vale winger drove a fierce shot which Lewington touched but couldn't keep out of the bottom right-hand corner of his net. An expected avalanche of pressure from Vale failed to materialise and substitute Dwight Gayle should have won it for the visitors when he blazed high and wide with the goal at his mercy.