Avery, playing for the fourth time since returning from a 10-game absence due to a shoulder injury, slapped home a power-play goal 9:40 into the second period to give the Rangers a 3-1 lead. Dubinsky stretched the advantage to three at 2:48 of the third.
Scott Gomez netted his fourth, scoring off an assist from Avery, and Michal Rozsival added a power-play goal in the first period for the Rangers, who won their fifth consecutive game at home and snapped a streak of four straight two-goal games.
New York, which started the season with a 5-2 win over Florida, scored more than two goals for just the third time in 12 games, but have six victories in eight games.
The Rangers, 8-2 at home and 0-5-1 away from Madison Square Garden, will open a four-game road trip Saturday against Toronto.
Pittsburgh captain Sidney Crosby earned his 15th assist to extend his career-best point streak to 15 games.
LIGHTNING 5, HURRICANES 1
In Raleigh, N.C., Shane O'Brien scored two goals to lead Tampa Bay to its first road victory of the season.
Vincent Lecavalier, Filip Kuba and Vaclav Prospal each scored for the Lightning, who were 0-7 away from home this season. Prospal and Michel Ouellet added two early assists as Tampa Bay won consecutive games following a six-game losing streak.
Johan Holmqvist finished with 40 saves, including an impressive diving glove stop on Erik Cole's short putaway during one of Carolina's few power plays midway through the second period.
DEVILS 4, FLYERS 1
In Newark, N.J., Veteran Dainius Zubrus and rookie Rodney Pelley scored their first goals of the season in a three-goal flurry to help New Jersey beat Philadelphia.
Zubrus and Brian Gionta scored within 18 seconds of each other in the first period after spotting Philadelphia a 1-0 lead, and Zubrus scored again late in the period to give New Jersey a 3-1 lead.
Pelley's first NHL goal early in the second period sent Flyers goaltender Martin Biron to the bench.
CAPITALS 4, SENATORS 1
In Ottawa, Viktor Kozlov had a goal and two assists in the second period and Washington ended Ottawa's team record eight-game winning streak.
Olaf Kolzig made 27 saves for Washington, which had lost four in a row and 10 of its last 12 (1-11-1), including a 2-1 overtime loss Tuesday in Atlanta.
Alexander Ovechkin scored his 10th goal with 3:45 left in the third to restore Washington's three-goal lead after Kozlov, Nicklas Backstrom and Tomas Fleischman each scored in the second period to put the Capitals up 3-0.