Andreas Lämmel, the conservative candidate, beat his Social Democrat rival by a slim margin, giving Ms Merkel's party an additional seat in parliament. The Dresden poll was held two weeks after Germany's inconclusive general election because of the death of a candidate.
Mr Lämmel's victory raises the CDU's parliamentary lead over the SPD to four seats and puts Mrs Merkel in a stronger position to claim the chancellor's job in a coalition government. Volker Kauder, general secretary of the Christian Democrats, said the result had strengthened both the party and Ms Merkel. "I am looking to rational forces in the SPD to recognise this and free themselves from Gerhard Schröder," he said.
The general election on 18 September left no party with enough votes to form a coalition with its preferred partner. Ms Merkel's conservatives have been trying to form a grand coalition with Chancellor Schröder's party as a way out of the political stalemate.
Who should become the next chancellor has emerged as the the main stumbling block to an agreement. Neither side has renounced its claim to the job.
Ms Merkel has insisted that her party will not enter formal coalition talks with the SPD unless Mr Schröder agrees to step down. She insists that, as her party won more seats than the SPD, she has a moral right to the job in any grand coalition.
Her stance was echoed by senior Christian Democrats at the weekend. Norbert Röttgen, the party's parliamentary business manager, warned: "If we allow the Social Democrats to decide who is number one, it will take us years to recover from the damage."
Mr Schröder has doggedly refused to step down. He claims that he is more popular than Ms Merkel and that, as the election produced a de facto left-wing majority comprised of his SPD, the Greens and the recently formed radical Left Party his remaining chancellor would more accurately reflect voters' wishes. However, Mr Schröder has categorically ruled out forming a coalition with the Left Party.
The SPD rejected media reports that Mr Schröder would step down at a party meeting today, a public holiday in Germany to celebrate reunification. " We are for Gerhard Schröder becoming chancellor, but we will talk about the whole constellation in our negotiations," SPD chief Franz Muentefering said in words less forceful than before.