The Seagulls claimed the scalp of the Magpies for the second season in succession, recording a 2-0 victory in their third round clash at the Amex Stadium courtesy of goals from Andrea Orlandi and Will Hoskins.
Magpies manager Alan Pardew, who saw star striker Demba Ba depart for Chelsea on Friday and had a number of key players out through injury, made five changes to his starting XI and witnessed a limp display from the Premier League side.
The dismissal of captain Shola Ameobi in the second half added to their woes and Poyet said: "We realised the opportunity and we were lucky that Newcastle couldn't bring their best players.
"We didn't create too many chances today but we were a good team to play against a Premier League team."
Brighton scorer Will Hoskins believes their victory over Newcastle reflects the ambition shown by the Championship side.
Hoskins, signed from Bristol Rovers in 2011, netted the second goal late on after Orlandi had put the Seagulls ahead before the break with a clever finish.
Hoskins said: "We did what we set out to do, play our way and not get fazed. It's a good result for us. This is why I signed here, great ambition and the club's fantastic."
Orlandi's opener was well taken with a deft touch putting his side ahead and he added: "It was a really nice goal, I was pleased with that.
"I don't know how I did it to be fair but it was a nice goal. We all played really well. It's great for us."