Columbus scored the only goal of the first period, while shorthanded, when T.J. Brodie's blind pass up the middle was intercepted by Letestu, who made a nice move to deposit his eighth of the season behind Miikka Kiprusoff.
The Blue Jackets took advantage of several more Calgary miscues in the second period scoring four times, all at even strength, to open up a 5-3 lead.
It was a big win for Columbus, who entered the night 0-2-1 on the first three stops of its four-game road trip, as they climb to within one point of the St. Louis Blues for the eighth and final playoff spot in the Western Conference.
The Blue Jackets, who were red-hot with points in 12 straight games prior to heading out on the road, are back home Sunday against the Anaheim Ducks.
The first game of the post-Jarome Iginla era in Calgary saw the Flames play a poor defensive game in which sloppy play and giveaways lead to many of the Columbus goals.
The Flames all-time leading scorer was traded to Pittsburgh on Wednesday night in exchange for a 2013 first-round draft pick and U.S. college prospects Ben Hanowski and Kenneth Agostino.
Nonetheless, many of the fans in the crowd, announced as a sell-out of 19,289, wore red jerseys with Iginla's name on the back.
The appreciative fans then stood up and gave a standing ovation in the first period when a stirring video tribute to Iginla was played on the jumbotron. While the fans rose to their feet, officials didn't immediately drop the puck as all the Flames players were banging their sticks against the boards.
Tied 1-1, Columbus took the lead for good at 8:44 of the second when Umberger poked the puck past Jay Bouwmeester at the Blue Jackets' blue-line and led a two-on-one rush that was finished off nicely by Prospal for an easy one-timer.
Less than 90 seconds later on another odd-man rush, Calvert looked off Jack Johnson and shot instead, ripping a shot over Kiprusoff's glove to put Columbus ahead 3-1.
Calgary, to their credit, kept answering back against Steve Mason and were within a goal at 4-3 when Johansen stripped the puck from Matt Stajan at centre and beat Kiprusoff on a breakaway with 1:05 left in the period.
The four goals in the second came on eight shots and resulted in Kiprusoff watching the third period from the bench as coach Bob Hartley inserted Joey MacDonald for the final 20 minutes. Bouwmeester was also on the ice for all four of Columbus goals in the second. He finished a game-low minus-4.
Calgary outshot Columbus 33-25.
Notes: Columbus rookie D Dalton Prout was a game-high plus-4 while also getting in a fight with McGrattan ... Stajan picked up his 200th NHL assist ... McGrattan's goal was his first in 75 games, dating back to Oct. 16, 2009 ... Calgary D Anton Babchuk was a healthy scratch for the 23rd time this season. He's played just two games ... Flames played their final home game before Wednesday's NHL trade deadline. Calgary has one more game before then, Monday night in Edmonton.