Van Riemsdyk opened the scoring with an unassisted goal off a Washington turnover 10 minutes into the game and added his sixth of the season shortly thereafter for the Leafs, who broke a two-game skid.
Korbinian Holzer added his first NHL goal and Phil Kessel collected a pair of assists in the victory.
Ben Scrivens turned aside 24-of-26 shots for Toronto.
Marcus Johansson and Mike Ribeiro each found the net for the reeling Capitals, who dropped to 2-7-1 to begin the season.
Michal Neuvirth stopped 18-of-21 shot as Washington fell for the fourth time in five contests.