The Senators should consider themselves lucky to walk away with the two points after showing little emotion through most of the first two periods.
Gonchar's winner came after Erik Karlsson wove through the Capitals' defence on a man advantage before finding his defence partner, who fired a one-timer that deflected off Hendricks in front and past Neuvirth with Ottawa's Chris Neil screening on the play.
Trailing 2-0 late in the second period and looking for a spark, Senators coach Paul MacLean chose to double-shift the one line that showed any jump.
The gamble paid off as O'Brien, who celebrated his 24th birthday, cut the lead in half with 1:23 remaining in the period on a setup from Erik Conda and AHL callup Mika Zibanejad.
The Senators then quickly tied the game at 2:20 of the third. Kyle Turris won a faceoff in the offensive zone and pushed the puck ahead to Michalek, who scooped it passed a surprised Neuvirth.
Playing without Jason Spezza because of an undisclosed upper body injury, the Senators struggled and the Capitals owned much of the momentum in the first period and jumped out to a 2-0 lead.
Washington then made it 2-0 as Hendricks tipped Jay Beagle's shot home.
The Senators continued to struggle in the second and seemed incapable of breaking through the Capitals' forecheck as Washington pinned the home side in its own end for nearly two minutes, leading to a chorus of boos from the 19,267 on hand at Scotiabank Place.
The Senators are back in action Wednesday when they host the Montreal Canadiens, while the Capitals head to Toronto to take on the Maple Leafs on Thursday.
Notes: LW Kaspars Daugavins and D Andre Benoit were healthy scratches for the Senators. Apart from Spezza, D Jared Cowen (hip, out for season), D Mike Lundin (finger, two weeks) also missed out. ... C Brooks Laich (strained groin, out indefinitely), D Jack Hillen (upper body, out indefinitely) sat for Washington with injuries, while C Marcus Johansson, D Tom Poti and D Roman Hamrlik were healthy scratches.