They are designed to encourage landlords and hospitals to treat gay couples the same way as married ones.
However, the documents do not provide the same legal status as marriage.
BBC Tokyo correspondent Rupert Wingfield-Hayes says campaigners are welcoming "a small step forward in a deeply conservative country".
Younger people in Japan generally support gay rights but the prime minister has previously urged caution over changing the law.
Hiroko Masuhara, 37, and Koyuki Higashi, 30, were the first to register their partnership in Shibuya district.
Ms Higashi, an actress, told reporters she hoped the measure would spread across Japan and eventually lead to same-sex marriage being legalised.
Japan does not have any national legislation specifically banning discrimination against gay people. Couples complain of being blocked from tenancies and prevented from visiting loved ones in hospital.
"I hope the day will come soon when there will be equality in society," Ms Higashi said, according to the AFP news agency.
Shibuya district mayor Ken Hasebe, congratulated the couple.
"It took a long time to get to here," he said.
Another wealthy district of the capital, Setagaya, has also started to offer the certificates.
Setagaya's mayor Nobuto Hosaka issued five same-sex couples with the papers on Thursday afternoon, according to the Kyodo news agency.
Mobile phone networks NTT DoCoMo and KDDI have previously said they would offer their family discounts to same-sex couples who provide a certificate.

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