Fiji took the lead in the first half when Vanisha Kumar powered past the Solomons goalkeeper Margaret Kofela.
But the Solomons, which was playing in its first Nations Cup game since 2010, gained an equaliser before half-time.
Jemina David was tumbled over in the box by Unaisi Tuberi and a penalty kick was awarded. Solomons captain Ileen Pegi sent the goalkeeper the wrong way for the leveller.
Fiji gained the upper hand in the second half, with Mereoni Tora twice heading wide and Kumar forcing a fine reaction save from Kofela, but their foes held on for the point.