Splitting polygons with line grid in QGIS?
I would probably do the following:
Run the Lines to polygons tool:
Vector > Geometry Tools > Lines to polygons
Or the Polygonize tool (as suggested by @GaborFarkas):
Processing Toolbox > QGIS Geoalgorithms > Vector geometry tools > Polygonize
Run the Clip tool using the output from Step 1 as the Input layer and your polygon layer as the Clip layer:
Vector > Geoprocessing Tools > Clip