Modano made it all worth it for the Stars, who had allowed an NHL-high 23 goals in starting the season 1-3-1.
Girardi's clearing attempt in front of Rangers goalie Henrik Lundqvist went right to Modano in the left circle. He snapped off a shot that beat Lundqvist high 8:03 into the third for his third goal.
The Stars tied it 1-1 in the second on captain Brendan Morrow's power-play goal. Marty Turco made 28 saves.
Markus Naslund had given New York a 1-0 lead in the first during a man advantage. Lundqvist stopped 25 shots but appeared to fight the puck all night.
New York is 1-2-1 since jumping out to a 5-0 start. The Rangers, who played for the seventh time in 11 days, have scored one goal or fewer in three of four games - including a 1-0 shootout win over Toronto on Friday.
They were helped out when the Stars showed a lack of discipline in the opening minute, and the agitating Avery had nothing to do with it.
Just 25 seconds after the initial faceoff, Loui Eriksson was sent off for tripping. The stoppage brought a bit of excitement as Avery yapped at Lundqvist before Brandon Dubinsky, who occupied the stall next to Avery at the Rangers practice facility last season, stepped in to defend him.
Avery and Dubinsky appeared to jaw at each other during the pregame warmup, when Avery also yelled in the direction of Lundqvist and backup goalie Steve Valiquette.
The Rangers made the most of their first power-play chance just 33 seconds into the advantage. Michal Rozsival sent a drive hard off the end boards behind Turco, and the rebound came right to Naslund at the right post for an easy putback at 58 seconds.
Naslund, the Rangers' top forward pickup in the summer to replace Jaromir Jagr, netted his second goal and first since New York's opening game of the season in the Czech Republic.
The next stoppage of play didn't occur until 8:17 later in a fast-moving period that featured 17 shots, nine by Dallas.
A crescendo of boos toward Avery got louder each time he touched the puck in the final two minutes of the first.
Dallas got even on its first power play. The Stars put the pressure on and nearly scored when Modano struck Lundqvist on the mask with a shot he didn't see.
The Stars gathered the puck again, and Morrow netted his second of the season off a rebound of a shot by rookie sensation Fabian Brunnstrom. Morrow swiped the puck in while down on his left side 5:28 into the second period.
Brunnstrom burst into the spotlight last week when he became the third NHL player to score three goals in his league debut. He netted his fourth in three games on Saturday against Colorado. The setup for Morrow was the 23-year-old Swede's first NHL assist.
Notes: Henrik Lundqvist faced twin brother Joel, a Dallas center, for the third time in the NHL. Their sister was in attendance. ... The Rangers were 50-20-16 with Avery in the lineup and 9-13-2 when he didn't play. ... Naslund had four goals and three assists in four games against Dallas last season while with Vancouver. ... Modano has six points in six games.
