Greening opened the scoring in the second period for a struggling Ottawa power play that came in on an 0-for-24 slump.
With five Islanders, including DiPietro, around him, Greening jammed in a loose puck home for his second of the season at 6:01.
Benoit then added his goal from Silfverberg just 2:36 later.
The Islanders put on the pressure, but Anderson held off New York until Harmonic scored early in the third.
Islanders coach Jack Capuano said his team played a good game, especially after a 7-0 home loss to the Philadelphia Flyers on Monday.
"It's a loss but (the) guys obviously worked hard. We had a lot of chances, we stayed the course and kept to the game plan but their goaltender played well," Capuano said. "We had a good system review (Tuesday) morning. Obviously our game wasn't very good against Philadelphia and we just wanted to go out and work hard for one another and play with a lot of passion. I thought we did that. We did a lot of good things."
Notes: The Senators were 1 for 5 on the power play, while the Islanders went 0 for 4. ... Chris Neil reached 1,900 penalty minutes for his career, all with the Senators, in Monday's win at New Jersey ... Islanders defenseman Matt Carkner missed his homecoming Tuesday as he sat out the game with a groin injury. Carkner, a native of Winchester, just outside Ottawa, spent three seasons with the Senators before signing with the Islanders in the offseason. ... Matt Moulson has played in 262 consecutive games, which is the fourth-longest streak in Islanders' history.