Jordin Tootoo added an empty-net goal for the Predators, who snapped a four-game skid. Rinne was sharp, turning aside 10 shots in the first period, 13 in the second and 10 in the third for his 17th career shutout.
"We had a desperation in our game tonight," Rinne said after the Predators ended a four-game losing streak. "The penalty killing did a good job. The power play did a good job. All of the areas we have been practicing have improved."
After a seven-game winning streak, the Coyotes have lost two straight in regulation for the first time this season. Phoenix will try to avoid a three-game skid Wednesday night in Minnesota.
"This was a big game for us, but that is part of what the NHL is," captain Shane Doan said. "You get lots of chances to respond, and we have a great chance tomorrow night."
Weber scored the game's first goal at 3:26 of the opening period. The Nashville captain had the puck in the right circle when his attempted pass into the low slot caromed back to him, and he sent a wrist shot into an open net.
O'Reilly doubled the lead at 12:48 of the second. With the Predators on a 5-on-3 power play, Steve Sullivan (two assists) sent a pass from the left of the Phoenix goal across the crease to O'Reilly, who had an open net of his own.
"We just went through our options," O'Reilly said. "Sullivan did all the work, and I had an empty net there."
That two-man advantage was created when the Coyotes took three consecutive penalties in the middle of the second period. O'Reilly scored with 3 seconds left on a 5-on-3. It was his fifth goal of the season.
Nashville entered with the NHL's 27th-ranked power play, clicking at just 11.1 percent. However, the Predators have two power-play goals in two of their last three games.
"The 5-on-3 goal can really be a momentum changer, and for us, it really gave us some breathing room," Nashville coach Barry Trotz said. "I thought Sully made two great passes on the power play and that's what clicked tonight."
Ilya Bryzgalov faced just 22 Nashville shots. It was the Russian's first road regulation loss of the season. He is 5-1-2 away from home.
"They played well on the power play, especially in the zone," Bryzgalov said. "They move the puck side to side really well and always had somebody in front of me."
Phoenix has won just twice in its last 11 games at Nashville.
"We had some guys really play hard, but in the end, you can't take penalties like that and expect to be successful," Coyotes coach Dave Tippett said. "That's something that will be addressed."
Material from wire services and team media was used in this report.