Jordin Tootoo also scored for the Predators, who snapped a four-game skid.
After a seven-game winning streak, the Coyotes have lost two straight in regulation for the first time this season.
"This was a big game for us, but that is part of what the NHL is," Phoenix captain Shane Doan said. "You get lots of chances to respond, and we have a great chance tomorrow night."
The Coyotes will try to bounce back Wednesday night in Minnesota.
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 five-on-three power play, Steve Sullivan 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. The Predators first had a five-on-three for 9 seconds, followed by another five-on-three for 50 seconds. O'Reilly's goal came with three seconds left on that five-on-three.
"We had some guys really play hard, but in the end, you can't take penalties like that and expect to be successful," Phoenix coach Dave Tippett said. "That's something that will be addressed."
The goal was O'Reilly's fifth of the season. Sullivan assisted on both of Nashville's power-play goals.
Nashville entered with the league's 27th-ranked power play, clicking at just 11.1 percent. But now, the Predators have two power-play goals in two of their last three games.
"The five-on-three 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."
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. "The penalty killing did a good job. The power play did a good job. All of the areas we have been practicing have improved."
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."
Tootoo's empty-net goal in the final minute completed the scoring.
Phoenix has won just twice in its last 11 games at Nashville.
Weber's goal was his 175th career point. He is tied with Marek Zidlicky for second in Nashville history for points by a defenseman. ... Nashville center David Legwand missed his sixth consecutive game with a lower-body injury. ... The Coyotes had not allowed a power-play goal in their previous three games. ... Phoenix left winger Scottie Upshall played his first game in Nashville since being traded from the Predators to the Philadelphia Flyers in the Peter Forsberg deal in February 2007.