The money went to a small group of social media experts that included Jonathon Morgan, the chief executive of New Knowledge, a cybersecurity firm.
They created a Facebook page intended to look like the work of conservative Alabamians, and used it to try to split Republicans and promote a conservative write-in candidate to take votes from Mr. Moore.
They also used thousands of Twitter accounts to make it appear as if automated Russian bot accounts were following and supporting Mr. Moore, according to an internal report on the project. The apparent Russian support for Mr. Moore drew broad news media coverage.
Democratic political strategists say the small Alabama operation — which accounts for a minuscule share of the $51 million spent in the contest — was carried out as a debate about tactics intensified within the party.
Democrats had been shocked to learn of Russia’s stealth influence campaign to damage Hillary Clinton and promote Donald J. Trump in the 2016 presidential race. But at least a few Democrats thought their party could not shun such tactics entirely if others were going to continue to use them.