A sharply divided D.C. City COuncil approved and sent to the mayor yesterday a $1.3 billion 1979 city operating budget that includes provisions for - but does not assure - up to $15 million in property tax relief.

The council concluded nearly two months of deliberation over the city's proposed spending request for the fiscal year beginning next October when it approved the budget by a slim 7-to-6 vote. The COuncil also passed a $130.2 million capital improvements budget.

At the same time, the Council postponed consideration of a controversial permanent rent control measure until later this month. The bill would extend the city's 3-year-old rent control system until 1980. It was rescheduled for debate at a committee of the whole session to be held within the next two weeks.

The Council already had extended the city's present rent control system until Jan. 31 through an emergency measure enacted last month.

Council Chairman Sterling Tucker said after yesterday's session that despite the close vote, he believed there never was any serious possiblity that the spending package would be disapproved. Tucker acknowledged that should Mayor Walter E. Washington veto portions of the budget bill - as he is almost certain to do - there may not be nine votes on the Council to override some vetoes.

The major now has two weeks to act on the budget and he can approve or reject individual items in it. The Council then has 30 days in which to override the mayor's decision before the city's budget is sent to the White House, which then forwards it to Congress.

As approved yesterday, the operating budget is $15.1 million less than the amount proposed by the mayor Sept. 12, and lowers by 54 positions the number of regular employees the city will be permitted to hire. The mayor had set the employee ceiling at 36,190; the Council shaved that to 36,136.

The Council has tentatively set aside the $15.1 million in a "reserve fund" with no use specified. Several Council members have said that they hope the money can be used to grant property tax relief to city home owners.

As proposed by the mayor, and approved by the Council, the budget does not increase the city's tax rate, but does assume a 23 per cent increase in the assessed value of all residential and commercial properties over the next two years. No other tax increases or higher user fees are included.

The Council also concurred with the mayor's recommendation to include in the budget only enough money to finance a 3 per cent pay increase for regular city employees. If a larger increase is granted, more funds would have to be sought.

The close Council vote stemmed from reservations individual Council members said they had regarding the budget approved yesterday and the process by which it was developed.

Council members Marion Barry and Polly Schackleton for example, disagreed with the mayor's revenue projections and some methods used to balance the budget. John Wilson opposed the process by which the budget was put together, contending that it did not properly set priorities and contained "luxuries that we can't afford."