The group has said on Twitter that the deadly clashes took place over the past 48 hours around a Houthi checkpoint in Sanaa's Arhab district.
It added that the three militants with whom contact was lost during the clashes have likely been killed.
This is the first time for Ansar al-Sharia to announce its presence in Arhab, which lies close to Sanaa's airport as well as several foreign embassies and government building.
The announcement comes after armed tribesmen in Sanaa said they would not confront Houthi militants, which took control of the capital in late September.
In recent months, the Houthis have emerged as a major power, changing the fractious country's political map.
The Shiite group's political ambitions have pitted them against Al-Qaeda, which remains influential in central and southern Yemen, and local Sunni tribes.
Thus, Yemen continues to see daily political violence which has left hundreds of civilians, tribesmen, army troops and Shiite Houthis dead so far.
The Arab country has been hard hit by political turmoil since a popular uprising that erupted in 2011 toppled longstanding president Ali Abdullah Saleh one year later.