However, Iran's ISNA student news agency said reports that Iran had fired missiles at Israel after the ceasefire took effect were false.
The developments on Tuesday raised early doubts about the ceasefire, intended to end 12 days of war.
Israeli Defence Minister Israel Katz said in a statement he had ordered the military to "continue high-intensity operations targeting regime assets and terror infrastructure in Tehran" in light of "Iran's blatant violation of the ceasefire declared by the President of the United States".
Hours earlier, Trump had posted on Truth Social: "THE CEASEFIRE IS NOW IN EFFECT. PLEASE DO NOT VIOLATE IT!"
Israeli Prime Minister Benjamin Netanyahu had confirmed the ceasefire, saying Israel had achieved the goals it had set in launching its June 13 surprise attack on Iran, to destroy its nuclear program and missile capabilities.
"Israel thanks President Trump and the United States for their support in defence and their participation in eliminating the Iranian nuclear threat," Netanyahu had said.
Iran's Foreign Minister Abbas Araqchi had said that Iran would halt its retaliatory strikes provided that Israel stopped attacking as of 4am (10.30am) in Tehran.
Global stock markets surged and oil prices tumbled on Tuesday after the announcement of the ceasefire, in the hope it heralded a resolution of the war just two days after the United States joined it by hitting Iranian nuclear sites with huge bunker-busting bombs.
Trump's declaration on Monday of what he called a complete ceasefire to end the war came after a sharp escalation of the conflict since Sunday, when the United States bombed Iranian nuclear sites, prompting Iran to retaliate by firing missiles at a US base in Qatar on Monday.
Both Israel and Iran reported strikes on their countries in the hours leading up to the ceasefire, with Israel's military saying six waves of missiles were launched by Iran and Israel's national ambulance service said four people were killed in Beersheba.
In Iran, the deputy governor of the northern province of Gilan said four residential units were destroyed in a "terrorist attack" on Tuesday morning, killing nine people.
A senior White House official said Trump had brokered the ceasefire deal in a call with Netanyahu and Israel had agreed so long as Iran did not launch further attacks.
An Iranian official earlier confirmed Tehran had agreed to a ceasefire, but the country's foreign minister said there would be no cessation of hostilities unless Israel stopped its attacks.
Israel, joined by the United States at the weekend, carried out attacks on Iran's nuclear facilities after alleging Tehran was getting close to obtaining a nuclear weapon.
Iran denies ever having a nuclear weapons program but Supreme Leader Ali Khamenei has said that if it wanted to, world leaders "wouldn't be able to stop us".
Qatari Prime Minister Sheikh Mohammed bin Abdulrahman Al Thani secured Tehran's agreement during a call with Iranian officials, an official briefed on the negotiations said.
Earlier on Monday, Trump said he would encourage Israel to proceed towards peace after dismissing Iran's attack on an American air base that caused no injuries and thanking Tehran for early notice of the strikes.
He said Iran fired 14 missiles, calling it "a very weak response, which we expected, and have very effectively countered".
Iran's attack came after US bombers dropped 13.6-tonne bunker-buster bombs on Iranian underground nuclear facilities at the weekend, joining Israel's air war.
The Trump administration maintains its aim was solely to destroy Iran's nuclear program, not to open a wider war.
with AP and PA
