Manchester City looked set to suffer Wembley heartbreak for the second time in less than 12 months on Sunday when Sunderland took a deserved lead through Fabio Borini.
But second-half goals from Yaya Toure, Samir Nasri and Jesus Navas denied the Black Cats a morale-boosting win.
Sunderland's players were dejected when they came off the Wembley pitch. Vito Mannone, for one, was almost inconsolable when he left the stadium.
But given that Sunderland are currently mired in a relegation battle, Poyet knows he cannot sit back and allow the players to wonder what might have been.
"It's up to me to make sure we use this defeat in the right way," the Sunderland manager said.
"I don't like excuses, I won't give the players the opportunity to have an excuses because we lost today. No chance."
Poyet, whose team are one point from safety, admits his management skills will be put to the test over the next couple of weeks as he tries to refocus the players' minds on moving out of the bottom three in the Barclays Premier League.
The former Brighton manager is sure that his players are good enough to escape the drop though.
"If we do this every week, I'm sure we'll stay in the league," he added.
"I need to see who takes my words now and shows me they are okay after this because we are all different.
"Maybe playing in the FA Cup (versus Hull) next week helps us, and then we have the (league) game against Crystal Palace which we must win, for sure."
While Poyet spoke about trying to get his team out of the relegation zone, his opposite number Manuel Pellegrini was dreaming about pulling off an incredible quadruple after lifting his first trophy in English football.
City are still in the FA Cup and they lie second in the Premier League. They are two goals down in their Champions League tie with Barcelona, but Pellegrini hopes he can guide his team to glory in all three remaining competitions.
"We are the only club that has the chance to continue trying to win all the competitions and this gives us a lot of confidence," the City manager said.
"We wanted this trophy but I don't think that anybody can think that is enough.
"I don't think for top players or important clubs, you are satisfied with one trophy.
"We know it won't be easy, but we are going to try."