The results show, with half of the election districts counted, that Williams is ahead with 926 votes to Falk's 894.
The pair ran to succeed Michael Griffin.
Meanwhile, incumbent Councilmen Shawn Rogan and Charles Cook, both running as Republicans, are slated to be re-elected after running unopposed. Unofficial results show that Rogan has 1,426 votes, while Cook has 1,416.
Town Clerk Antoinette Kopeck, who ran unopposed, is heading to re-election with 1,521 votes, unofficial results show.
Highway Supervisor Russell Goff, who also ran without opposition, is heading to re-election and has 1,490 votes.
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