By Sarah Marsh and Matt Spetalnick

WASHINGTON (Reuters) - Cuba's top diplomat warned the United States on Tuesday against taking hasty decisions over alleged incidents that have harmed U.S. embassy staff in Havana and urged its authorities to cooperate on the investigation into the mysterious affair.

Cuban Foreign Minister Bruno Rodriguez had called for Tuesday's meeting with U.S. Secretary of State Rex Tillerson in Washington to discuss the case, which has been threatening the already fragile detente between the two former Cold War foes.

The highest level meeting to take place between the two countries since U.S. President Donald Trump took office came 10 days after Tillerson said the United States was considering closing its recently re-opened embassy in Havana.

"The Foreign Minister reaffirmed that the investigation to resolve this matter is still in progress," the Cuban Foreign Ministry said in a statement, noting that "effective cooperation" of U.S. authorities was essential.

"It would be regrettable that a matter of this nature is politicized and that hasty decisions not supported by conclusive evidence and investigation results are taken."

The meeting "took place in a respectful ambiance", the Cuban Foreign Ministry said.

The U.S. State Department said the conversation was "firm and frank" and Tillerson "conveyed the gravity of the situation and underscored the Cuban authorities’ obligations to protect Embassy staff."

Washington earlier this year expelled two Cuban diplomats over the alleged incidents, which it says caused symptoms from hearing loss to nausea in U.S. embassy staff and their families in Havana, although it has so far not laid blame on Cuba.

Cuba reiterated on Tuesday that it "has never perpetrated nor will it ever perpetrate attacks of any kind against diplomats," and said it had implemented additional measures to protect U.S. personnel since they reported the incidents.

The case has brought simmering tensions between the two countries since Trump took office to the boil.

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Trump, who in June vowed to partially roll back the detente with Cuba agreed by his Democratic predecessor, President Barack Obama, called the Communist-run nation "corrupt and destabilizing" in his address to the United Nations General Assembly last week.

He said he would not lift the U.S. trade embargo on the Caribbean island until it made "fundamental reforms." Cuba described his comments as "unacceptable and meddling."

Cuba's investigation had uncovered "no evidence so far of the cause or the origin of the health disorders reported by the U.S. diplomats," the country's Foreign Ministry said.

Experts agree it is hard to see how any attacks could have been carried out or what the motivation could be.

Theories abound, from surveillance technology gone awry to a sophisticated acoustic weapon in the hands of Cuban-American exiles or third-party state actors such as Russia, Iran or North Korea, but most flounder.

Audiologists, for example, have raised doubt over the possibility of whether any sonic weapon exists that can be used covertly to bring about the range of symptoms mentioned by diplomats.

(Additional reporting by Yara Bayoumy and Patricia Zengerle in Washington; Writing by Sarah Marsh in Havana; Editing by Leslie Adler)