The Senate completed congressional action yesterday on the District of Columbia budget, granting the city a record $336 million federal payment but rejecting a proposed 16 percent pay increase for City Administrator Elijah B. Rogers.

Senate action sent the $1.9 billion D.C. appropriation bill for the fiscal year that began Oct. 1 to the president for his signature. By taking the action, Congress removed the District budget from a controversy between Congress and the White House over the continued level of funding for government agencies.

Unless the lawmakers reach agreement on a continuing resolution today, federal agencies will have no authority to spend money beyond midnight. Until yesterday, the District would have been covered by that restriction.

A House-Senate conference committee, reconciling differences between the two chambers' versions of the city budget, agreed on a Senate-passed provision giving Mayor Marion Barry the right to raise the pay of his top aide, Rogers, from the current maximum of $50,800 to $61,400. The full House voted on Wednesday, however, to reject that provision. A section providing funds to run the city's new lottery board remained intact.

Rogers said yesterday that "we told the Senate to go ahead and approve [the House version]. We don't want our budget held up for that one [salary] item."

As for reports that Rogers may be lured away by higher pay in other cities, he said, "I've told everybody all along I'll be here through the mayor's first term. I'll still be here." Barry's term ends in January 1983. He is expected to seek reelection.

The House District Committee, on a separate legislative track, is considering a bill that would authorize higher pay for Rogers. If enacted, it would overrule yesterday's appropriations action.