Defeat for Coventry, managed by former England midfielder Frank Lampard, ended the Sky Blues' run of six straight league wins and left them just three points clear of second-placed Middlesbrough.
Victory took Wrexham, owned by the Hollywood duo of Ryan Reynolds and Rob McElhenney, into the top half of the table.
Coventry went ahead in the 22nd minute when Ephron Mason-Clark swept home a first-time finish on the half-volley from Tatsuhiro Sakamoto's excellent cross from the right after beating the offside trap.
But Wrexham hit back in the second half to the delight of home fans at the Racecourse Ground.
The Welsh club, who have enjoyed three successive promotions under their celebrity owners, were level on the hour mark.
Issa Kabore fed Josh Windass down the right, and his cross was turned home by the sliding Moore with a right-foot finish.
Nine minutes later, a throw down the right wing led to another assist from Windass, with his ball nodded in by Moore.
The Wales striker then took his impressive goal tally this season to nine in 16 appearances, seven minutes from time, when, with Windass the provider again, Moore drilled a low, left-foot shot into the far corner of the net.
Coventry gave themselves a lifeline in the 88th minute when Japan forward Sakamoto let fly with a superb left-foot shot that curled into the far corner.
But Wrexham, beaten by Welsh rivals Cardiff in the League Cup in midweek, saw the game out.