"Over 750 people have been rescued since 5:00 am (0300 GMT)," Yazan al-Saadi, MSF's regional communications officer said in Beirut.
He said a fourth rescue operation by MSF's Bourbon Argos search-and-rescue vessel was currently underway.
In videos sent to AFP by MSF, hundreds of people in brightly-coloured clothes are sprawled out on the ship, some hugging and clapping.
Children can be heard crying in the background.
"There are multiple nationalities... Thankfully, everyone is in good health," said MSF spokesman Sami al-Subaihi, who is seen aboard the ship in the video.
The Bourbon Argos has 26 people on board, including medical specialists.
"Once rescued, MSF treats the refugees on the ship in case of any cases of dehydration or other medical issues," Saadi told AFP.
"Then we take them to our centres either in Italy or Greece," he added.
Yesterday, 102 migrants were rescued in the Mediterranean by Libyan authorities, and seven other bodies were recovered.
Another 124 migrants were detained by Libyan coastguard officials yesterday as they were preparing to cross to Europe.
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