Palace have not won a match in 2023 and were beaten 1-0 by Brighton in the Premier League on Wednesday, their third defeat in a row.
They are 12th in the table but are only three points above 18th-placed Bournemouth, who have a game in hand.
“It is with enormous regret that this difficult decision has been made,” chairman Steve Parish said.
“Ultimately, results in recent months have placed us in a precarious league position and we felt a change is necessary to give us the best chance of retaining Premier League status.”
Frenchman Vieira, 46, was appointed before the start of the 2021-22 season and led the Eagles to a 12th-placed finish, earning praise for an attacking style.
However, this season they have scored just 21 goals in 27 games – only Wolves, Everton and bottom-of-the-table Southampton have fewer.
Before their defeat by Brighton, Palace had gone three consecutive games without a shot on target.
No Premier League team has won fewer points in 2023 than Palace, who have taken five from a possible 33.
Three members of Vieira’s coaching staff – Osian Roberts, Kristian Wilson and Said Aigoun – have also left the club and the process of appointing a new manager is under way.
Just five points separate the Premier League’s bottom nine clubs heading into the final months of the season.
Palace travel to leaders Arsenal, Vieira’s former club, on Sunday (14:00 GMT).
Their final 10 games of the season will see them face all eight of their relegation rivals.
Parish added: “Patrick’s impact since joining us in the summer of 2021 has been significant, and he is held in the highest regard by myself, and all of his colleagues.
“He led the team to a Wembley FA Cup semi-final and respectable 12th-placed finish last season playing some exciting football, which was a challenging and crucial campaign for the club given the changes we made to the squad prior to his arrival.
“Patrick has given his all to the club, and we all thank him and his team for their service.”
Vieira’s departure means there are no black managers in charge of a club in the top flight of men’s and women’s football in England.
Newcastle United striker Callum Wilson said this week he believes some black footballers in England may be put off going into management by their experiences of racism as players.
In the wider Premier League, only six managers have been in charge for more than three years and nine have been in charge for fewer than 12 months. The league average stands at two years and five days.
There have been nine managerial sackings during the 2022-23 campaign – one fewer than last season’s total, which equalled the Premier League record.
‘The threat of relegation proved too much’ – analysis
It was felt that Vieira was fighting for his job after the loss to Brighton on Wednesday, but he may have been given the Arsenal game as one last chance to show he could turn around a season that is turning into a relegation battle.
Ultimately, that has not happened.
Palace have not won since a New Year’s Eve victory over Bournemouth. They conceded with the last kick of the game against Brentford on 18 February, which cost them a win, and have had fixtures against the majority of the top six to deal with.
Goalscoring has been a huge issue – Palace have not found the net for the past four matches and have not scored twice in a game since December.
It is not yet clear who the club will bring in to replace Vieira but they will need to get the team scoring and quickly. The Eagles do not concede a lot but without scoring themselves, the threat of relegation gets bigger with every week that passes.