MARJAH, Afghanistan—U.S. and Afghan troops invaded this Taliban-held town early Saturday, launching the main thrust of the largest coalition offensive since 2001, a test of whether America's surge strategy can rescue the faltering war effort.

British, Afghan and U.S. troops were poised to begin an even larger thrust to take over one of the last Taliban-controlled bastions in the volatile Helmand River Valley. The offensive in Marjah began with small teams dropping into the town around midnight to kill or capture insurgent...