Mane headed in Trent Alexander-Arnold's cross to open the scoring on nine minutes before producing a brilliant backheeled finish to double the lead 11 minutes later.
Divock Origi fired in to make it 3-0 in the 66th minute before substitute Andre Gray forced Alisson into two saves in three minutes as Watford pushed for a reply.
Van Dijk took the match away from the Hornets with two headed goals in three minutes.
The first came from Alexander-Arnold's cross for the right-back's third assist of the match, the second from Andy Robertson's centre.
The result puts Liverpool on 69 points, one point above Manchester City.
Watford drop to eighth with 40 points.