LONDON — Prime Minister Theresa May of Britain faced down hard-line critics Wednesday and won the support of a jittery and divided cabinet for a plan to quit the European Union, preserving her push to avert an economically damaging rupture with the bloc in March.

For Mrs. May, frequently criticized as wooden and lacking in strategic thinking, the victory represented a rare validation of her leadership. It also provided a glimmer of light at the end of the Brexit tunnel.

“The choices before us were difficult,” Mrs. May said in a brief statement in front of her offices at 10 Downing Street. “But the collective decision of the cabinet was that the government should agree to the draft withdrawal agreement and the outline political declaration.”

Despite fears that cabinet ministers might resign in protest, possibly threatening her government, none did so immediately. Nevertheless, Mrs. May acknowledged publicly that around five hours of talks had produced “impassioned” debate. That description seemed to corroborate media reports that as many as 10 cabinet ministers had expressed reservations before the plan was agreed upon without a vote.