"Both parties have agreed that it is in the best interests of the club to make this change," Middlesbrough, three points from safety and without a win in 10 matches, said in a statement.
The club, promoted last season after a seven-year absence from the top flight, thanked Karanka for his work over the past three-and-a-half years.
Middlesbrough are 19th in the Premier League, with a game in hand over Hull City who are two points above them but three adrift of 17th placed Crystal Palace.
With only 19 goals from 27 matches, and just four wins all season, 'Boro have scored the fewest goals in the top flight despite keeping a tight defence.
They were knocked out of the FA Cup after a 2-0 loss to Manchester City in last Saturday's quarter-final.
Agnew will take charge for Sunday's big home clash against Jose Mourinho's Manchester United, who are on an unbeaten 17 match league run but will be without suspended top scorer Zlatan Ibrahimovic.
CRITICAL OF SUPPORTERS
"I'd like to thank Middlesbrough for a wonderful opportunity and the players, staff and all the people at the club who I have worked with," Karanka, a former assistant to Mourinho at Real Madrid, said on the club website (www.mfc.co.uk).
"I'd also like to thank the fans for their support," added the north-eastern club's first non-British manager, who had said last week that he was confident he still had the backing of chairman Steve Gibson.
"This club will always hold a special place for me and I wish everyone connected with Middlesbrough Football Club the very best for the future."
Karanka had been critical of the club's supporters earlier this season, when they called openly for a change of tactics, and had bristled at the club's lack of activity in the January transfer window.
There have also been spats with high-profile players such as former England winger Stewart Downing, who has been dropped for recent games.
"We need to improve the team, and the club knew a month and a half ago the players that I wanted," Karanka had said at the end of January.
"Teams in our position are signing players for 14 million pounds ($17.17 million) -- we are signing players that didn't play in the (second-tier) Championship."
($1 = 0.8156 pounds)
(Reporting by Alan Baldwin, editing by Sudipto Ganguly)
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