John, who had already seen a penalty saved, netted in the 19th minute before Morgan levelled the score shortly after the half-hour mark.
The Blades' comeback was completed with little more than five minutes to half-time when Gray forced the ball home from close range.
But it was the hosts who had made all the early running as Stephen Hughes chanced his arm with a drive which was saved by Ian Bennett, the goalkeeper on loan from Birmingham.
Next, Louis Carey sparked panic in the Blades' penalty area when a booming cross was dispatched from the right wing to the back post where Danny Cullip tangled with Dele Adebola.
The Sky Blues' demands for a penalty were waved away by referee Mike Dean, but, on the quarter-hour mark, a second round of appeals met with success after Cullip tangled with Adie Williams.
But John's side-footed effort was saved low to his left by Bennett.
Trinidad and Tobago international John made amends in the 19th minute though, after a Hughes free-kick was nodded on by Adebola for Gary McSheffrey to provide a tight cutback from the byline.
And, after turning in the six-yard box, John made no mistake.
At the other end, Luke Steele saved a diving header from Andy Liddell.
But it was the hosts who continued to run the show as Adebola had a shot blocked and then John hit the post.
However, the initiative was to swing markedly after the half-hour when the Blades won a corner on their left wing.
Liddell crossed but, with Steele marooned in no man's land after failing to claim the ball, Morgan punished Coventry for their ineffective defending.
The visitors went on to take the lead with six minutes of the half remaining after Phil Jagielka cut in from the right to supply Gray, who lunged to slide the ball home at Steele's near post.
Steele was earning early second-half pats on the back from team-mates as he denied Liddell at the back post from point-blank range before Coventry swarmed upfield in search of an equaliser.
But, on this occasion, McSheffrey could only screw his shot across the six-yard box and wide.
Florent Laville was punished after hauling down Danny Cadamarteri just outside the Coventry box and, from the subsequent free-kick, Jon Harley forced Steele into an ungainly save at the foot of his left post.
From the resultant Liddell corner, Morgan went close to scoring his second goal of the afternoon when his powerful header rebounded off the inside of Steele's woodwork and into the goalkeeper's arms.
With 15 minutes to go, a carbon copy of the same routine ended with Morgan heading over before Liddell tested Steele with a long-range free-kick.