The Capitals came from behind twice to force overtime with Alex Ovechkin scoring both goals for the visitors but it would be the Islanders who would clinch the win with Tavares flicking the puck past Washington goaltender Braden Holtby with 24 seconds left in the first extra period.
The win was the Islanders' third from their past four matches that went beyond the standard three periods, after shootout triumphs against the Philadelphia Flyers and Pittsburgh Penguins.
Anders Lee and Travis Hamonic scored the earlier goals for the Islanders.
Tavares & Co's victory on Wednesday kept the Islanders on track with Metropolitan division leaders Pittsburgh, who overcame the Toronto Maple Leafs 4-3 in overtime.
The Penguins struck twice in the opening period but a brace from Leo Komarov saw Toronto draw level in the second stanza before Blake Comeau's second goal edged Pittsburgh ahead again.
Tyler Bozak of the Maple Leafs notched the fourth goal of the second period with less than four minutes before the break and after a goalless third, Comeau completed his hat-trick in overtime to hand the home side victory.
Pittsburgh lead the Islanders (both 32 points) as they have played one less game than the New York-based team, while Washington are third ahead of the New York Rangers (both 22) due to their superior head-to-head record.
The Rangers lost 4-3 at the Tampa Bay Lightning, who moved within a point of Atlantic division leaders the Montreal Canadiens.
Stanley Cup champions the Los Angeles Kings smashed the Minnesota Wild 4-0 and the Detroit Red Wings cruised past the Flyers 5-2.
In other results, the Winnipeg Jets won 2-1 at the Buffalo Sabres, the Florida Panthers edged past the Carolina Hurricanes 1-0, the Chicago Blackhawks defeated the Colorado Avalanche 3-2 and the Calgary Flames triumphed 2-0 against the San Jose Sharks.