It was a performance that will give some encouragement to Wales boss Gemma Grainger in only her second game in charge.
Gemma Evans forced a fine save out of Danish keeper Lene Christensen in the early stages before the Danes hit back to take the lead. Chelsea forward Harder's 65th international goal drew her level with Merete Pedersen's national record.
But Wales were far from outclassed by the world's number 14-ranked team and came close to equalising before the break through Fishlock and Ceri Holland.
Christensen then saved superbly from Angharad James as Wales were unfortunate not to reach the interval on level terms.
Fishlock finally made the breakthrough before Wales withstood some heavy Danish pressure in the dying minutes to earn a creditable draw.