The visitors held the upper hand in the first-half and the only surprise was it took 34 minutes before they went ahead through Callum Wilson's ninth goal of the season.
The home side drew level in the 40th minute with their first attempt. Chris Lines swung over a free-kick from the right and Tom Pope rose superbly to glance a 10-yard header inside visiting keeper Joe Murphy's right-hand post.
But Coventry regained the lead in sensational style three minutes later.
Franck Moussa surged forward from the centre circle and crashed an unstoppable right-footed shot into the top left-hand corner of Neal's net from 30 yards.
A double substitution sparked a massive improvement from Vale in the second-half as Ben Williamson, twice,and Pope squandered great chances to level.
But it was now time for the Coventry rearguard to buckle and Vale skipper Doug Loft slammed in a low 10-yard shot for a 59th-minute equaliser after the visitors had failed to deal with another Lines' free-kick.
There were further chances at both ends before Birchall's right foot squeezed a deserved winner, low inside Murphy's left-hand post from four yards after Williamson and Pope had both had shots blocked.