Though Mike Jackson's men started the final day outside the bottom three on goal difference, they would be usurped by Leeds United, who won 2-1 away at Brentford.
Callum Wilson was on the scoresheet twice for the Magpies — the first from the spot after Nathan Collins' inexplicable handball — as Burnley's six successive seasons in the top flight came to an end.
He said: "I’m sorry that Burnley have gone but they’ve got to hold their heads high because the way they’ve performed the past six or seven years they’ve shown the character, they’ve shown how good they are as a team. They’ll be back, they’ve got a fantastic team and a very good coach.
"Obviously it’s a disappointing moment. They’re going to feel down, of course they are, but that’s the emotion of the game. But they’ll come back strong.
"I know a lot of the players, I know a lot of the characters. They’ve got to stick together. You need that team spirit and they’ve got that here. I know they have. Even when I’ve not been there for many years I still speak to all the players, so they’ll be back for sure."
The ex-Atletico Madrid full back, who collected a La Liga winner's medal last term, made 170 league appearances for Burnley after being brought to the club from Manchester City by Eddie Howe.
And he can empathise with the Clarets having endured relegation under Sean Dyche seven years ago, prior to his move to Tottenham Hotspur. Despite having a hand in the club's return to the Championship, he said: "I’m a professional player and I’m not at Burnley anymore, I’m at Newcastle, and obviously I wanted to win the game.
"But this is football, I’ve been relegated with Burnley before and I know good mates who have played on the opposite side.
"I’ve been there before and I know how it feels. It’s not a nice feeling, but they’ve got the squad here. They’ve got the characters who’ve been down and come back up two or three times now. They’ve shown great character this season, especially the last few weeks."
The World Cup semi-final goal-scorer, who featured in England's Euro 2020 final with Italy, also admitted that he was shocked to hear the news of Sean Dyche's departure last month.
The Clarets' 2011-12 Player of the Year, who made 120 of his 185 appearances in all competitions under Dyche, is still in touch with his former boss, who he has maintained a close relationship with.
Trippier said: "I’m still in touch with Sean Dyche. We’ve had a great relationship since I left.
"I’m not at Burnley every day so I don’t know what’s happening behind the scenes but I was training and I heard that they’d parted ways. Obviously for everybody in football it was a bit of a shock.
"Dychey’s a fantastic manager but the new gaffer’s come in and done a fantastic job. The results that they got many people wouldn’t have predicted that so he deserves huge credit."