The border facility is set to boost trade and investment between the two countries, Saudi Customs governor Ahmad Al-Qabbani said.
Subsequently the television Al Massirah of the Houthis confirmed the launch but said several missiles were meant to strike the Saudi capital and not one.
Residents said they heard two powerful explosions hit the building, which is located near a hotel, a bank and shops, and not far from the central bank.
Yemen's Iran-backed Houthi rebels have fired ballistic missiles into the Kingdom in the past, which have been intercepted by Riyadh. It also fired missiles in Riyadh's Dry Port and other "economic targets" in the city.
He later said in a separate statement that the Houthis had "failed to launch a short-range Badr-type rocket" toward the Saudi province of Najran but it fell in a non populated desert area.
"There will be more salvos until this enemy is deterred, understands the meaning of the Yemeni might and ceases its crimes", he pointed out.
The cross-border attack by the Iran-allied rebels, known as Houthis, came amid mounting regional tensions after U.S. President Donald Trump chose to withdraw from the landmark 2015 nuclear agreement with Tehran. But there are fears that the decision could exacerbate the conflict in Yemen and other regional flashpoints.
Saudi Arabia and its allies intervened in Yemen in 2015 with the aim of pushing back the rebels and restoring the internationally recognised government to power.
United States ally Saudi Arabia has reiterated that it will build a nuclear bomb if its regional enemy Iran ramps up development of a weapons program.
Saudi authorities later claimed that its air defense systems had intercepted the missiles in the skies over the capital. Col. Malki said these hostile actions by Houthis backed by Iran prove the continuing involvement of the Iranian regime in supporting the militias with qualitative capabilities in a clear violation of United Nations resolutions. It has instructed citizens upon hearing the alert to seek shelter in secure locations and avoid "areas prone to air strikes and missile strikes".