Splitting polygons with line grid in QGIS?

I would probably do the following:

1. 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
   

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2. 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