"You are taking office at a difficult time in which you face a great responsibility," Merkel said in the message released by the chancellery in Berlin.
She expressed the hope that his coalition government would "further strengthen and deepen the traditionally good and deep friendship between our peoples.
"For your future work as prime minister, I wish you much strength and success," she added.
Her spokesman said a day earlier that Germany, the paymaster for eurozone bailout packages, would listen closely to how the new Greek government sees its "future reform course and the fulfilment of its commitments".
"In our view it is important for the new government to take action to foster Greece's continued economic recovery," Steffen Seibert told reporters.
"That also means Greece sticking to its previous commitments."
Tsipras, whose radical left-wing Syria party won a stunning victory in Sunday's polls, has vowed to reverse many of the belt-tightening measures that Greece's creditors insisted on in return for its 240-billion-euro bailout deal.
Merkel has pushed for swingeing budget cuts and strict economic reforms in response to the eurozone debt crisis.
Many in Greece and other stricken countries blame her for compounding the suffering of their people and choking off economic recovery.
But Germany and its allies argue that only fiscal discipline will put Europe on a sustainable path to growth.
Merkel's no-debt-reductions stance was backed by EU Commission chief Jean-Claude Juncker, who warned on Monday that Greece cannot expect any reduction of its debt commitments.
"There is no urgent need for action" on Greece's debt, Juncker told German television station ARD, adding that a reduction of the debt "is not on the radar".
"I don't think there's a majority in the Eurogroup... for a reduction of the debt," he said, referring to the eurozone's finance ministers.
The head of the Eurogroup, Dutch Finance Minister Jeroen Dijsselbloem, warned Greece earlier that the problems facing the Greek economy had "not disappeared" with Syriza's victory.
Juncker, who spoke Monday with Syriza leader Alexis Tsipras to congratulate him on becoming prime minister, said the Mediterranean country would have to "move towards the commitments" it had taken previously in respect of its massive bailouts by the EU and the International Monetary Fund.
Juncker vowed to work with Athens "in the interest of all" to consolidate "the considerable progress of the past few years."
"We mustn't turn the clock back now but work on this basis to enable further progress - helping Greece to have more growth, more jobs, more investment."
SEE ALSO: 'Stick to the plan': Germany to Greece