The Alexandria City Council last night approved construction of a controversial new high-rise apartment building for the elderly poor.

The 6-to-0 vote was a victory for the city's Redevelopment and Housing Authority, which had argued that the facility, to be called Fairfax House, is a crucial part of a plan designed to keep the agency from bankruptcy.

In past weeks, the planned seven-story, 90-unit project at Pendleton and North Royal streets in Old Town faced stiff opposition from opponents who argued that the land for the site is too valuable for use as public housing when it could be developed to yield high property tax revenue to the city.

Some members of the Old Town Civic Association scrutinized zoning codes and met with federal housing officials in an attempt to stop the new construction.

But at a public hearing last Saturday, developer John K. Freeman told the council that to postpone the project meant losing government financing that would subsidize rents for the one-bedroom apartments that would otherwise rent for about $500 a month. Fairfax House is scheduled to be completed by May 1983 and will house elderly residents from the John Roberts Homes, a public housing project that is to be sold to keep the housing agency solvent.