In dramatic scenes, demonstrators swarmed near the building, as Congress members were escorted out by police.
A joint session of Congress has been counting and confirming electoral college votes.
The proceedings are usually brief and ceremonial but Republican lawmakers are objecting to some results.
Any attempts to block Mr Biden's victory are almost certain to fail.
For days Mr Trump has also been putting pressure on Vice-President Mike Pence, who is presiding over the session, to block certification of the result.
But in a letter to Congress on Wednesday, Mr Pence said that he had no "unilateral authority to decide which electoral votes should be counted".
Nevertheless, Mr Trump is making a final stand against the certification of his defeat.
The protesters arrived at the Capitol from a "Save America Rally", where Mr Trump had urged them to support those legislators opposing Mr Biden's confirmation.
Mr Trump has refused to concede the 3 November election, repeatedly alleging fraud without providing any evidence.
On Wednesday, he said again: "We will never give up. We will never concede."
He has also tried to throw doubt on the integrity of Tuesday's Senate run-off votes in the southern, traditionally Republican, state of Georgia. Projections by US TV networks suggest the Democrats have won one of the seats and are neck-and-neck for the second seat.
If the Democrats win both they will gain effective control of the Senate - something that will help Mr Biden push forward his agenda after he is inaugurated as president on 20 January.