Is mowing an effective way to control weeds?

For the most part, no.  Mowing can actually increase shoot density of some plants with a creeping root system.  Research has shown that several mowings on Canada thistle will introduce stress and set up the plant for a fall-applied herbicide.

After being cut, most weeds will re-grow shoots from the reserves in the roots back into the upper plant in order to make flowers and seeds.  Certain plants can even create multiple stocks or heads where one was cut off, thus increasing the seed count.  When controlling a bi-annual seed production plant, like Musk thistle, the root should be cut one to two inches below soil surface prior to flowering, which will kill the plant.  Preventing flowering and seed formation is critical for success for plants that only spread by seed.

Some studies have shown that after years of repeated mowing at the right time can eventually exhaust a plants reserve in the root system.  However this is not recommended unless you plan on mowing multiple times a season.