Greening scored twice and Craig Anderson made 36 saves in the Senators' 4-1 victory over the San Jose Sharks on Thursday night.
Kyle Turris and Erik Karlsson also scored for Ottawa. The Senators have won six of their last seven games and have points in 11 of 12. They won for the first time in San Jose in eight years.
"We're very confident right now," Greening said. "This shows we're a good road team. To come in here and win is a testament to our resiliency."
Anderson won his fifth straight road game and is 7-0-2 in his last nine road games.
"They came out pretty hard at the start," Anderson said. "They have a history of coming out hard. We just had to weather the storm and get through the first five minutes without getting too bloodied up. If we hang around long enough, we always have a chance."
The Senators are 12-2-2 since acquiring Turris in a trade with Phoenix.
Brad Winchester scored for the Sharks, who won the previous five meetings with Ottawa.
"We had some scoring chances early," Sharks captain Joe Thornton said. "If we convert a couple of those, things could have changed. They probably had six or seven chances and they made good on a lot of them."
Winchester opened the scoring 3:10 into the game after Andrew Desjardins won the puck along the board and feed him for the one-time shot.
"We weren't good enough with the forecheck and our power play wasn't good," San Jose's Logan Couture said. "We need to be better. You can have as many chances as you want but this is about results. You need to score goals."
Turris tied it 7 minutes later.
"Andy stood on his head again," Turris said. "It seems like every game he's making saves no one else can make."
Karlsson's goal early in the second period gave the Senators the lead for good. Turris started the counterattack for Ottawa.
Greening scored a power-play goal late in the second period on a rebound. He added a goal in the third period on a give-and-go with Jason Spezza.
The Sharks were 9-1-1 in their previous 11 games.
NOTES: Senators D Sergei Gonchar appeared in his 1,100th game. ... Ottawa D Filip Kuba had his 100th point with the Senators. ... Desjardins returned to the lineup after missing two games with an upper-body injury. ... Sharks D Ryane Clowe missed his second straight game because of a head injury. ... Sharks D Dan Boyle has eight assists in his last six games. ... Greening had his second career multi-goal game.