The broadcaster said it will do all it can to ensure uninterrupted coverage of the Games after members of the National Union of Journalists (NUJ), Bectu and Unite voted in favour of the industrial action on July 23, the day of the opening ceremony.
BBC staff will strike from 11.59am until 11.59pm, affecting all parts of the corporation, followed by a work-to-rule from midnight on July 24.
The move comes in protest at a pay offer of £650 for those earning less than £50,000 a year and £500 for employees on more than £50,000. The NUJ wants to see staff awarded a better pay rise and says capping executive pay at £150,000 would free up money to do that.
Michelle Stanistreet, general secretary of the union, said: "The decisive turnout and result clearly demonstrate that journalists across the BBC are not prepared to put up with paltry pay deals any longer, while those running the corporation continue to enjoy their lavish salary and perks.
"Members are clear that they are prepared to take sustained action in this dispute and will name further dates if the dispute is not resolved. It's time for the BBC to re-evaluate its position and to resolve this dispute by negotiating a sensible and fair pay rise for staff who work so hard, for modest salaries."
The decision has prompted anger from some parties, with the Tories describing the move as unnecessary.
Scottish Conservative sports spokeswoman Liz Smith MSP said: "The news of this strike is extremely disappointing, especially because it is clearly deliberately timed to coincide with the launch of the Commonwealth Games.
"There is absolutely no need for this strike to go ahead. When athletes have given up so much to train and prepare for the opportunity of a lifetime, the various unions are preparing to strike, which has the potential to cause completely unnecessary unrest just when the Games are beginning."
A spokesman for the BBC said: "We will do all we can to bring our audience uninterrupted coverage of the Commonwealth Games. In the meantime, we will continue to speak to the unions in an attempt to resolve this dispute.
"However, we have already made an improved offer and we are mindful that across the BBC we need to make significant savings and deliver more for less."
A spokeswoman for Glasgow 2014 and Games Partners, including the Scottish Government, said: "We understand that the BBC are doing all they can to bring full coverage of the Glasgow 2014 Opening Ceremony to audiences."