Boston mayor Thomas Menino on Wednesday described the magazine's cover as "a disgrace".
While Rolling Stone defended its decision, drugstore chain CVS Caremark Corp refused to sell it.
"It's a total disgrace, that cover of Rolling Stone," Menino told reporters at the opening of a rail station.
"It should have been about survivors or first responders. Why are we glorifying a guy who created mayhem in the city of Boston? I am going to be in touch with the publishers and tell them how I feel about it."
Tsarnaev is the survivor of a pair of brothers accused of carrying out the worst mass-casualty attack on US soil since September 11 2001 killing three people and injuring more than 260 at the Boston Marathon on April 15 with a pair of homemade pressure-cooker bombs.
The August issue of the magazine depicts Tsarnaev, with long, shaggy hair and sporting a light beard and moustache, over the headline: "The bomber: How a popular, promising student was failed by his family, fell into radical Islam and became a monster."
'Out of state'
Tsarnaev (19) looks thinner and younger in the photo than he appeared last week in a Boston federal court to face charges related to the bombing, which carry the threat of execution.
Massachusetts Governor Deval Patrick was more restrained in his comments.
"I haven't read it, but I understand the substance of the article is not objectionable, it's apparently pretty good reporting," Patrick said. "But the cover is out of taste, I think."
The article, which Rolling Stone posted to its website on Wednesday, reveals a few new details about Tsarnaev, including that he once told a high school friend he believed terrorist attacks could be justified and he "took his religion seriously", according to a friend cited in the article.
In a statement, the magazine's editors defended their decision to feature Tsarnaev on a cover that has depicted music legends ranging from Bob Dylan to Jay-Z, as well as actors and other celebrities, over its 45-year history.
"The cover story we are publishing this week falls within the traditions of journalism," the magazine's editors said.
"The fact that Dzhokhar Tsarnaev is young, and in the same age group as many of our readers, makes it all the more important for us to examine the complexities of this issue."
Social media rage
The decision to put Tsarnaev on the cover drew a wave of outrage on social media, including Twitter.
CVS Caremark said on its Twitter feed it would not carry the issue "out of respect for the victims and their loved ones".
Dzhokhar Tsarnaev and his 26-year-old brother plotted the marathon attacks months in advance, authorities charge, traveling to New Hampshire to buy fireworks that they used in building the bombs.
Three days after the attack, the FBI unveiled video stills of the two near the finish line, in the hope that members of the public would be able to identify them.
That prompted the pair to try to flee the city.
Prosecutors say the men first killed a Massachusetts Institute of Technology police officer in an unsuccessful effort to steal his gun, then engaged in a gun battle with police that ended when Dzhokhar ran over his older brother, contributing to his death.
The younger Tsarnaev's escape led to a day-long manhunt that ended with his arrest late on April 19.
Tsarnaev appeared in court for the first time last week, and pleaded not guilty to all charges in a 30-count indictment.
He faces the possibility of the death penalty if convicted. – Reuters