The Blackhawks outshot the Predators 29-17 and generated ample chances despite playing their second game without Patrick Kane, the team's leading scorer when he was injured against St. Louis on Wednesday. The star forward is on injured reserve with a lower-body injury and could miss the rest of the regular season.
Chicago fell six points behind Western Conference and Central Division leader St. Louis, which defeated Pittsburgh 1-0 earlier on Sunday.
The Blackhawks' Corey Crawford made 15 saves.
Rinne was sharp and in his old form in his 31st career shutout, his first since March 12, 2013.
The two-time Vezina Trophy finalist had surgery on his left hip in May 2013 and recovered in time to play in nine of the Predators' first 10 games to start the season. But an infection in the hip required him to undergo a second surgery and he missed 51 games.
Rinne has started nine of 11 games since returning in a 3-1 loss to Pittsburgh on March 4.
Nashville defenceman Seth Jones suffered an upper-body injury and didn't return for the second period. Forward Patric Hornqvist, tied for second on the Predators with 38 points, missed the game and returned to Nashville to be with his wife following the birth of the couple's first child.
The Blackhawks outshot the Predators 11-6 in the first and dominated much of the period, but Nashville led 1-0 after 20 minutes.
Fisher opened the scoring on a deflection with 1:22 remaining in the first when he slipped in unchecked from the left side of the net. Ryan Ellis shot through a screen from the right point, and Fisher jammed in front and tipped the puck past Crawford.
The Blackhawks came out strong in the second, helped in part by a four-minute power play resulting from Nashville's Eric Nystrom penalty for roughing. But Rinne stopped several close-in chances and limited rebounds to keep the Predators ahead.
Bourque made it 2-0 with 2:25 left in the period on a rising shot from the left circle that sailed over Crawford's right shoulder as he dropped to the ice.
In the third, Rinne came through when tested and managed the puck conservatively, often smothering it to eliminate Chicago rebounds and chances.
Notes: Before the game, Blackhawks coach Joel Quenneville said there's "a good chance" top prospect Teuvo Teravainen could make his NHL debut on Tuesday against Dallas. Teravainen, Chicago's first-round draft pick in 2012, joined the Blackhawks on Friday and has been skating with the team. ... F Matt Carey, 22, signed by Chicago as a free agent on Thursday following one season at St. Lawrence University, also has joined the Blackhawks and could play down the stretch. ... Chicago F Brandon Saad returned on Sunday after missing three games with an upper-body injury. F Bryan Bickell and D Michal Rozsival remained out. ... Sitting out with injuries for Nashville were F Patrick Eaves and F Paul Gaustad. D Michael Del Zotto was a healthy scratch.