Cofie's superb second-half strike cancelled out the lead given to the Alex by Luke Murphy's penalty on the stroke of half-time.
The visitors almost went ahead in the fifth minute when Mark Ellis met Murphy's free-kick with a firm header but teenage goalkeeper Fabian Spiess palmed the ball away.
However, Crewe deservedly went into the break ahead after Gavin Mahon clumsily brought down Ajay Leitch-Smith inside the box after 45 minutes and Murphy dispatched the spot-kick with aplomb.
County came out revitalised in the second half and looked to have equalised when a sublime, dipping shot from 30 yards by Alan Judge hammered against the crossbar.
But the home side did not have to wait long to level when, in the 57th minute, a mistake by Adam Dugdale gave the ball to loan signing Cofie just outside the area and the 20-year-old planted a superb right-foot shot into the bottom corner of the net.
Murphy almost won it for Crewe with 13 minutes remaining when his free-kick was acrobatically tipped over by Spiess.