The home side required a miracle to stay up as Norwich needed to beat Arsenal, and see West Brom lose to Stoke in addition to a massive 17-goal swing.
But their keeper John Ruddy, hoping to be in Roy Hodgson's England squad, had to make three good saves - two from Olivier Giroud and one from Lukas Podolski - to just keep it level in the first period.
Ramsey opened the scoring on 53 minutes with a sublime volley from Giroud's pass before Jenkinson sealed the win nine minutes later with his first goal for Arsenal.
In an added bonus for Arsene Wenger's side, England midfielder Jack Wilshere and Abou Diaby also made welcome returns from the bench from long-term injuries.
Norwich, who sacked manager Chris Hughton with just five matches left, must now regroup for life in the Sky Bet Championship along with Cardiff and Fulham.
Wilshere made a return to the squad following a fractured foot suffered on international duty against Denmark in March, while with fourth place already secure and perhaps one eye on next weekend's FA Cup final, Lukasz Fabianski started in goal.
Caretaker Norwich boss Neil Adams found no place in the match-day squad for club-record signing Ricky van Wolfswinkel, who managed just one goal and along with captain Sebastien Bassong - another conspicuous by his absence - looks set to leave in the summer.
With nothing other than pride to play for, Norwich, who actually had a decent home record this season, started brightly.
On 17 minutes, winger Nathan Redmond - one of the few summer buys to have performed consistently well - cut in from the left and drilled an angled shot from 20 yards which Fabianski palmed behind.
At the other end, a clever backheel from Giroud in the left side of the penalty area created space for Podolski, but goalkeeper Ruddy was out quickly to make a smart save.
The Norwich No.1 then produced a great stop after a neat one-two put Giroud clear, but the forward's chipped drive was pushed over at point-blank range.
Arsenal were finally finding some rhythm, as another quick interchange on the edge of the box teed up Ramsey, whose snap-shot was blocked by Michael Turner.
Ruddy was again called into action five minutes from half-time when he tipped over a fine curling 20-yard effort from Giroud.
But the keeper was finally beaten eight minutes after the restart by a moment of pure quality.
Giroud held the ball up on the edge of the Norwich penalty area, before chipping a cross to the back post. Ramsey dropped a yard deep, before volleying the ball up over Ruddy into the top-right corner for a strike of the highest technical order.
With the game, and ultimately their season, gone, Norwich handed a first-team debut to Jamar Loza, the 19-year-old forward just back from spells at Leyton Orient and Southend.
Arsenal were 2-0 ahead on 62 minutes when full-back Kieran Gibbs carried the ball to the left touchline, before cutting it back into a crowded box.
Jenkinson prodded it home before the life-long Arsenal fan wheeled away to the travelling support in wild celebration of his first goal for the club.
Wilshere then replaced Ramsey to give him a run-out ahead of the FA Cup final and a likely summer trip to Brazil.
There was also a welcome late cameo for Diaby, the France midfielder recovered from a serious knee injury to play in his first match since March 2013, with his name reverberating around Carrow Road as the Arsenal fans paid tribute.
Fabianski saved bravely at the feet of Robert Snodgrass, the Norwich player of the season in a forgettable campaign, as Arsenal, already assured of a Champions League qualifying place, closed out a fifth straight league win.