But senior American officials who have worked closely with the Saudis in recent years to help them improve targeting procedures said that while the additional training was important, it would be effective only if Defense and State Department officials monitor the program closely.
“This training package sets an important precedent to focus on preventing civilian casualties,” said Larry L. Lewis, a former senior official at the State Department who visited Saudi Arabia five times in 2015 and 2016 to help the country’s air force improve its targeting procedures and investigations. “But the follow-through is critical. Those things are necessary but not sufficient to help them solve their problems.”
Saudi Arabia has faced mounting international pressure to find a face-saving way to justify a two-year campaign in Yemen that has damaged its image abroad as military errors have exposed shortcomings in the Saudi armed forces.
In addition to the thousands of people who have been killed, many Yemenis have been pushed toward famine while extremist groups like Al Qaeda and the Islamic State have taken advantage of the chaos to step up their operations in the country.
In October, the coalition bombed a funeral reception in Sana, the capital, killing more than 100 people. The coalition later said the attack had been based on false information. That debacle prompted the Obama administration to block a transfer of precision munitions to the kingdom because of concerns about civilian casualties that administration officials attributed to poor targeting.
The Trump administration reversed that decision, arguing that the Saudis needed the precision-guided munitions to help avoid hitting civilians. The Saudis finalized a long-discussed training package and gave Mr. Tillerson the assurances he needed to help defend the sale on Capitol Hill.
Indeed, some supporters of the sale cited the training and assurances from the Saudis.
“I am aware of the concerns with Saudi Arabia’s engagement in Yemen, including in operations that have led to civilian casualties,” said Senator Mark Warner of Virginia, one of five Democrats to join most Republicans in Tuesday’s 53-47 Senate vote supporting the sale. “I share these concerns and believe that the Saudis have a responsibility to conduct their operations carefully — including engaging with the U.S. on increased training.”
This training for the Royal Saudi Air Force and other Saudi forces, which American officials said has started, includes subjects like human rights, flight training and how to avoid civilian casualties.
In addition, Mr. Jubeir promised in his letter, which was viewed by The New York Times, that Saudi Arabia would adhere to the international Law of Armed Conflict and expand the list of targets in Yemen that are off-limits to airstrikes to about 33,000.
Two senior American officials said that in many Saudi strikes supporting troops under fire and targeting so-called pop-up targets, or militants on the move, Saudi military planners were not regularly consulting the no-strike list, which includes sites like mosques and marketplaces.
The Saudis also agreed to observe stricter rules of engagement and consider in their targeting procedures specific estimates about potential harm to civilians and civilian buildings — a practice not fully integrated in the Saudi-led air campaign, American officials said.
Finally, the Saudis will allow American military advisers to sit in the Saudi air operations control center in Riyadh; previously, a tiny American military team was permitted to operate only from another office to coordinate the limited American logistical assistance to the campaign.
“We feel we have truly reset the relationship as a result of his visit,” Timothy A. Lenderking, a deputy assistant secretary of state focusing on the region, said in an interview.
Congressional opponents of the arms sales strongly disagreed. In a speech on the Senate floor before Tuesday’s vote, Senator Chuck Schumer of New York, the Democratic leader, said Riyadh’s support for Wahhabism, a strict Muslim sect that adheres closely to the Quran, was responsible for much of the radicalization of Muslim youths in the Middle East.
“Furthermore, the administration has not sufficiently ensured the Congress that these weapons won’t fall into the wrong hands,” Mr. Schumer said.
Human rights and humanitarian groups also criticized the sale — with the conditions — as ignoring even larger problems looming over Yemen.
“The steps that Saudi Arabia has reportedly agreed to take are irrelevant because they will not keep seven million people in Yemen from tipping into famine or stem the tide of cholera,” said Scott Paul, the senior humanitarian policy adviser for Oxfam America.