WASHINGTON—Congress stands on the verge of passing a $1.5 trillion tax cut and the largest structural overhaul of the tax system since 1986.

The bill sailed through the House Tuesday afternoon on a 227-203 vote with little drama, then a last-minute glitch forced Republican leaders to schedule another vote in the House Wednesday. Every Democrat who was present voted no, as did 12 Republicans, almost all from high-tax states.

The...