You can use the approach that I have used with these raster and grid vector layer of next image. First, select the element to clip with "Select features by area or single click" at the grid vector layer. At the attributes table of grid layer, click in "Show selected features" option (at the bottom) and next, at the record, click with right button of the mouse for selecting "Open Form". Copy to your favorite text editor the extent parameters in this order: X_MIN, Y_MAX, X_MAX, Y_MIN (see next image).
Open Clipper Tool in Raster -> Extraction -> Clipper. Select the name of output file, and drag with the mouse one arbitrary area on the map and click on the icon pencil for editing gdal_translate command. Change projwin parameters for the values at your text editor (copy/paste).
After OK, the clipped image it will have the same resolution of raster base; as it can be observed at next image.