Yandle knocked in a rebound 41 seconds into the game for his fourth point in two games and Boedker scored 30 seconds into the third by squeezing a puck between Calgary goalie Danny Taylor's pads in front of a standing-room only crowd at Jobing.com Arena.
That was enough for Smith, who was sharp all night for his 22nd career shutout and third this season.
Antoine Vermette and Raffi Torres each added goals in the third period, David Moss had two assists against his former team and Oliver Ekman-Larsson added two assists in Phoenix's sixth win in eight games.
Taylor stopped 33 shots in his first NHL start.
The Flames have played well lately despite their revolving door of goaltenders.
Starter Miikka Kirprusoff hasn't played since Feb. 5 because of a sprained knee and Calgary sent Leland Irving to Abbotsford of the AHL after he gave up two goals on four shots in 5-2 loss to St. Louis on Friday.
Joey MacDonald, picked up off waivers from Detroit last week, earned his first victory with Calgary after the Flames rallied with three goals in the third period for a 4-3 win over the Stars on Sunday night, giving them at least a point in three of four games.
Against the Coyotes, Flames coach Bob Hartley decided to go with Taylor, whose only previous NHL experience was one period with the Los Angeles Kings during the 2007-08 season.
Phoenix got to him right away, with Yandle punching in a rebound after a sharp-angle shot from Moss, a former Flames forward, caromed off Taylor into the slot.
The Coyotes beat Taylor again early in the third period when a shot by Ekman-Larsson bounced hard off the back boards to the other side of the goal. Taylor dropped to his knees as he scrambled around the crease, but Boedker's shot slipped between his legs and bounced off his right pad into the goal.
Vermette, who had an assist on Boedker's goal, scored later in period by swatting the puck in during a scramble in front, and Torres scored on a wrister from along the right boards.
That was more than enough for Phoenix.
The Coyotes have gotten back to their defence-first style after a rough start to the season, earning at least a point in eight of 10 games, including a pair of shutouts by Smith.
Smith was sharp again against the Flames, positionally sound and tracking the puck well as the Coyotes limited the chances in front of him in the first period.
He was just as good in the second when Calgary picked up the offensive pressure, turning away 13 shots, including a snatching glove save on a wrister by Lee Stempniak late.
Smith was still sharp in the third after Phoenix broke it open, sending the Coyotes into a four-day break with a solid all-around victory.
NOTES: Willie O'Ree, the first black player in NHL history, attended the game and received a loud ovation when he was interviewed on the video scoreboard. ... The Coyotes played without RW Radim Vrbata and D Derek Morris due to injuries. C Matthew Lombardi and D David Schlemko also are on injured reserve. ... Phoenix's Paul Bissonnette and Calgary's Tim Jackman had a short fight in the first period that the officials broke up when Jackman pulled Bissonnette's jersey over his head and kept punching.