Powell, 44, had been in talks with new owner Roland Duchatelet over a contract extension, but they have not agreed on a number of key issues in recent weeks.
Riga, 56, had been technical director of AC Milan's academy and is known to Duchatelet, who also owns Standard.
He will take charge for the first time against Huddersfield on Wednesday.
Belgian businessman Duchatelet, who also has links to four other European clubs, took over the Addicks at the start of January.
Duchatelet said there had been "good progress" in talks over a new deal for Powell but that the two parties "could not reach an agreement over the club's football strategy going forward".
Five of Charlton's six signings in January came from Duchatelet's other clubs, while they sold striker Yann Kermorgant and midfielder Dale Stephens to Championship rivals Bournemouth and Brighton respectively.
"The situation put a strain on the working relationship between Chris and the board," 67-year-old Duchatelet said in a statement on the Charlton website.
"Therefore I think it is best for all parties that we part ways at this stage."
Charlton have only won six of their 30 league games this season but, despite being adrift at the foot of the table, have four games in hand on Millwall, who are in 21st place.