Daniel James' superb 89th-minute strike looked to have salvaged a point for United, who fell behind to Jordan Ayew's first-half effort.
Luka Milivojevic's 69th-minute foul on Scott McTominay handed United a lifeline, but Marcus Rashford - who Ole Gunnar Solskjaer confirmed is one of two designated penalty takers following Paul Pogba's miss against Wolves - struck his spot-kick against the upright.
James' sensational finish seemed to have sounded a victory charge, yet De Gea failed to make what should have been a simple near-post save from Patrick van Aanholt's 93rd-minute attempt, as Palace claimed their first league win over United since 1991.
Not long after Rashford squandered an early chance, James went close to breaking the deadlock but the winger's close-range strike was kept out on the line.
United's profligacy proved costly as Victor Lindelof lost a header to Jeffrey Schlupp and Ayew kept his cool to slot beyond De Gea.
Luke Shaw's hamstring injury compounded United's woes, and the hosts should have been further behind when De Gea denied Wilfried Zaha from point-blank range.
Anthony Martial wanted a penalty 10 minutes after the restart when he appeared to be dragged down by Martin Kelly, but Paul Tierney dismissed United's appeals.
However, there was little doubt over Milivojevic's lunge on McTominay.
Rashford stepped up but his strike flew to safety off the inside of the left-hand post.
James looked to have sparred his team-mate's blushes with a wonderful effort, only for De Gea to gift Palace the win when he parried Van Aanholt's shot straight down and over the line.
What does it mean? United's penalty debate set to rumble on
Solskjaer defended Pogba's decision to take the penalty against Wolves, insisting the midfielder and Rashford are his two spot-kick specialists.
Rashford took the responsibility on this occasion but similarly fluffed his lines and questions over United's approach to penalties are now sure to linger, especially given the ramifications of their latest miss.
James proves a rare bright spot
Of United's three signings in the off season, James seemed the most likely to have to wait for his turn in the limelight. However, his second goal in as many Old Trafford appearances and other flashes of quality provided Solskjaer with the only real positive from a dismal day.
De Gea's difficult run continues
United's goalkeeper came under scrutiny at times last term and his latest mistake will do little to quieten the critics. For a player of De Gea's quality, Van Aanholt's shot should have proved comfortable to deal with.
Key Opta facts
- Manchester United have won one of their past nine matches across all competitions (D3 L5), scoring eight goals while conceding 16.
- Van Aanholt became only the second player to score a 90th minute winner against United in the Premier League - after Thierry Henry for Arsenal in January 2007 - and the first to do so at Old Trafford.
- Manchester United are only the second team to win a penalty in each of their first three matches of a Premier League season, after Chelsea in 2012-13. However, the Blues scored all three of theirs.
- Since the start of last season, Crystal Palace have earned 60 per cent of their points in the Premier League in away games (32/53), more than any other side.
- James is the seventh Manchester United player to score in both of his first two Premier League appearances at Old Trafford and the first since Zlatan Ibrahimovic in September 2016.
- Ayew's opener came from Palace's first shot and first touch in the opposition box.
United are back on the road for their next outing, taking on Southampton at St Mary's Stadium, while Palace host Colchester United in the EFL Cup on Tuesday.