Despite the presence of Ivory Coast forward Wilfried Zaha, Palace once again drew a blank as an 86th minute goal from Ashley Fletcher condemned Vieira to his second defeat.
Thomas Frank's side were in danger of defeat in the semi-final second leg after Arnaut Danjuma put Bournemouth 2-0 up on aggregate early in the first half.
Scottish Premiership: Former Bournemouth boss Eddie Howe favoured to replace Neil Lennon as Celtic manager
Eddie Howe has held discussions with members of the club's board, including major shareholder Irish financier Dermot Desmond.
Southampton have been immune to their league woes in the FA Cup this term and they booked a place in the Wembley semi-finals thanks to Nathan Redmond's double after Moussa Djenepo's opener.
Henry said: "The last year has been an extremely difficult one for me personally. Due to the worldwide pandemic, I was unable to see my children. Unfortunately this season the situation will be no different. The separation is too much of a strain for me and my kids."
A fine finish from Scotland midfielder Scott McTominay ensured Manchester United advanced to the last eight for a seventh successive season
The 29-year-old, who spent a season on loan at the Vitality Stadium in 2016/17, has been without a club since leaving West Ham in October.
Liverpool will face Arsenal twice at Anfield in three days next week after Jurgen Klopp's much-changed side hit Lincoln for seven with Takumi Minamino and Curtis Jones both scoring twice.
Manchester City sign Bournemouth defender Nathan Ake on five-year deal after Cherries drop to Championship
Manchester City agreed to a fee of 40 million pounds for Nathan Ake, who signed a five-year deal.
Howe was English league football's second longest-serving manager having begun his second spell at Bournemouth in October 2012 and leading it into the Premier League in 2015.
Premier League: Manchester United, Chelsea seal Champions League spots; Bournemouth, Watford relegated
While Ole Gunnar Solskjaer has overseen a remarkable turnaround for United in the second half of the season to finish in third, Leicester collapsed to throw away the chance of what seemed a certain top-four finish.
Premier League: $250M+ at stake as Aston Villa, Bournemouth and Watford fight to avoid relegation on final day
Norwich has already been relegated after one season back in the top-flight. Two of the Premier League clubs Aston Villa, Bournemouth or Watford on Sunday will join Norwich in dropping down.
Premier League: Aston Villa jump out of relegation zone after beating Arsenal; Manchester City thrash Watford
Aston Villa might be the most confident of the trio after beating Arsenal 1-0 on Tuesday to climb out of the bottom three, where the team from central England has been since 29 February.
Premier League: Harry Kane stars in Tottenham's 3-0 win over Leicester; Southampton hand Bournemouth 2-0 loss
Leicester City's chances of making it to the Champions League and Bournemouth's Premier League survival took a hit after their defeats of 3-0 and 2-0 to Tottenham Hotspur and Southampton respectively.
Premier League: Aston Villa boss Dean Smith says players ready to fight to win remaining two games and stay in top division
Villa appeared on course for a crucial victory at Everton on Thursday but were dealt a massive blow in their quest to remain in the top division when Theo Walcott headed in an 87th-minute equaliser to earn the hosts a 1-1 draw.
Premier League: Eddie Howe believes Bournemouth are 'more than capable' of getting result at Manchester City
Bournemouth remain in dire straights in 18th position, three points behind Watford and West Ham United who are just above the relegation zone.
Bournemouth travel to second-placed Manchester City on Wednesday and finish the campaign against Southampton and Everton.
Leicester City, who host Sheffield United on Thursday, have won just once in six league games since the season’s restart.
Premier League: Chelsea lose 3-0 to Sheffield United; Liverpool's fine home run comes to end with draw against Burnley
Given Chelsea lost at Sheffield United 3-0 on Saturday in a surprisingly one-side match, Liverpool likely wouldn't have been too perturbed.
Premier League: Tottenham boss Jose Mourinho wants access to referees to seek explanation for decisions
Mourinho preferred not to discuss Oliver during his news conference on Friday, but said officials should be forced to speak to managers to explain decisions.