your demands," Sharaf told the flag-waving supporters in the square, the centre of protests that toppled president Hosni Mubarak in February.
The premier was named by the country's new military rulers on Thursday to replace Ahmed Shafiq, who was opposed by protesters because he had been appointed by Mubarak in the dying days of his regime.
Sharaf, who was born in 1952 and served briefly as a transport minister under Mubarak, had joined protesters in Tahrir Square during the uprising to demand political and economic change.
And in a rare move for an Egyptian official, he appeared in Tahrir Square on Friday shortly before Muslim prayers, to be greeted by raucous cheers.
Some formed a human tunnel to allow him to walk through the crush of supporters.
Repeatedly interrupted by chants of "We are with you", Sharaf pleaded with the crowd for patience as he began the work of heading a transitional caretaker government.
"I beg you, you did something great and together we will do more," he said. "I have a heavy task and it will need patience."
Sharaf's appointment was likely to placate protesters, who had continued to rally in Tahrir Square to demand that the cabinet be purged of old regime elements including his predecessor Shafiq.
Mubarak, 82, stepped down on February 11 after three decades in power.
"This is a good step but we still have demands. We want the detainees to be released," Omar Ahmed said from Tahrir Square.
"We are ready to give him time, but we will keep pressing for our demands," said Ahmed Adel, another protester.
Scores of people were detained during the deadly protests that led to the overthrow of Mubarak. At least 384 people were killed and more than 6,000 injured in 18 days of riots.
Sharaf paid tribute to all those who participated in the uprising and saluted "the martyrs of the revolution, the thousands who were injured and the families of the victims."
The Supreme Council of the Armed Forces, which took power after Mubarak's resignation, has pledged to oversee a transition to a free democratic system.
Friday's protest, which had originally been organised to press for further change, turned into a massive celebration following news that Shafiq had resigned.
Sharaf's speech was interrupted once more when protesters chanted in unison "The people want the end of the state security" in reference to a branch of the interior ministry long accused by rights groups of crushing dissent through torture.
"I pray that Egypt will be a free country and that its security apparatus will serve the citizens," Sharaf said.
"Lift your head up high, you're Egyptian," he told the adoring crowd, the signature chant when Mubarak resigned.