Neither side could be separated at the end of 120 minutes and the tie was eventually settled on penalties, with Borre netting the decisive spot kick after substitute Aaron Ramsey saw his effort from 12 yards saved by Kevin Trapp just moments earlier.
A closely-fought contest was played out in the opening exchanges in Seville, but it was the Germans who began the final on the front foot, with Allan McGregor the busier of the two goalkeepers in the first 25 minutes.
Rangers grew into the game and created chances of their own with Aribo's curling effort on his favoured left-foot the closest that they came midway through the first half.
Aribo was the man who eventually made the breakthrough when he capitalised on a slip by Frankfurt defender Tuta, which allowed the makeshift striker to run through on goal and place his shot past Trapp into the bottom corner.
Frankfurt responded well and piled the pressure on Rangers before scoring the equaliser through Colombian striker Borre, who managed to get onto the end of an inviting cross from Filip Kostic to poke past McGregor from inside the six-yard box.
After the tie was forced into extra time, Rangers had a glorious opportunity to win in the 117th minute, but Trapp made a remarkable save to keep out Ryan Kent from just a few yards out.
Trapp then went on to make another vital save in the shootout, keeping out Ramsey's tame effort from 12 yards, before Borre rounded off five perfect spot kicks from Frankfurt to clinch their first European trophy since 1980.
12th min: The first opening of the final falls for Frankfurt, with Daichi Kamada finding space on the left side of the penalty area, but the Japan international delays his shot before Djibril Sow watches his strike from the edge of the area saved comfortably by Allan McGregor.
20th min: Ansgar Knauff cuts inside onto his left foot and gets a strike away from inside the area, but McGregor, at full stretch, makes a brilliant save, tipping the ball behind for a corner.
26th min: Scott Wright is wiped out on the edge of the area, but the referee plays the advantage as Joe Aribo swivels and curls a strike on his favoured left foot narrowly wide of the top-left corner.
32nd min: A bursting 70-yard run down the left flank from Filip Kostic results in the winger fizzing a low-driven strike across goal and wide of the far post.
36th min: Borna Barisic curls a free kick into the box from the left towards Lundstram, whose looping header is tipped over the crossbar by Kevin Trapp.
45 + 2 min: Aribo does well to retain possession on the right flank and eventually picks out Ryan Jack on the edge of the penalty box but he lashes a right-footed effort over the crossbar.
49th min: Jesper Lindstrom swivels on the edge of the area before watching his deflected strike fly narrowly wide of the left post.
54th min: Rangers' first chance of the second half falls to Ryan Kent but he fires a left-footed strike wide of the post after a quick counter-attack.
57th min: Rangers capitalise on a Frankfurt mistake to break the deadlock before the hour mark. Tuta fails to deal with a looping ball forward and slips, allowing Aribo to run through on goal, with the makeshift striker calmly slotting a left-footed strike past Trapp, sending the Gers fans behind the goal into pandemonium.
59th min: Frankfurt appeal for a handball inside the penalty area but Lindstrom's strike from a central area is blocked by the chest of Calvin Bassey.
67th min: A mistake from Connor Goldson allows Rode to play Kamada in behind Rangers' backline, but he can only dink his effort over the crossbar.
69th min: Frankfurt draw level courtesy of a poacher's finish from Rafael Borre. An inviting delivery by Kostic from the left flank is whipped in towards the near post, where Borre is waiting to pounce and poke the ball past McGregor.
76th min: Frankfurt substitute Jens Petter Hauge fires a shot high over the crossbar after some neat footwork on the edge of the area from Rode.
89th min: Rangers substitute Fashion Sakala gives the ball away in the middle of the pitch, before Kostic picks up the ball on the left flank and fires a cross-come-shot beyond Kamada and wide of the far post.
94th min: The first chance in extra time falls to Rangers substitute Scott Arfield, who leans back and lashes a 25-yard strike over the crossbar.
106th min: A powerful long-range effort from Barisic swerves towards goal but is saved well by Trapp before Frankfurt clear their lines.
112th min: Kent finds himself in space and decides to tests Trapp with a 25-year strike, but the German goalkeeper comfortably gets behind the shot.
117th min: A glorious chance for Rangers to win the match goes begging! Trapp makes a crucial save to keep out Kent from just a few yards out before Steven Davis blazes the rebound high over the crossbar.
120th min: James Tavernier curls a free kick up an over the wall towards the top-right corner, but Trapp is there to take it in his grasp.
RANGERS PEN ONE: Tavernier scores, the Gers skipper slots the ball into the bottom-left corner, sending Trapp the wrong way to put Rangers 1-0 up.
FRANKFURT PEN ONE: Christopher Lenz scores, the centre-back calmly places his effort in the bottom corner to make it 1-1.
RANGERS PEN TWO: Steven Davis scores, the veteran midfielder steps up and fires a brilliant strike into the roof of the net to put Rangers 2-1 up.
FRANKFURT PEN TWO: Ajdin Hrustic scores, the Austrian looks nervous when placing the ball down on the spot, but he sends McGregor the wrong way to make it 2-2.
RANGERS PEN THREE: Scott Arfield scores, Trapp goes the right way but is unable to get fingertips on the strike which flies into the top-left corner, putting Rangers 3-2 ahead.
FRANKFURT PEN THREE: Daichi Kamada scores, McGregor dives the right way but Japanese playmaker's powerful strike rifles in off the post to make it 3-3.
RANGERS PEN FOUR: Aaron Ramsey misses, a tame effort straight down the middle from the Welshman is saved by Trapp, keeping the score at 3-3.
FRANKFURT PEN FOUR: Filip Kostic scores, as he did with Kamada's spot kick, McGregor dives the right way but is unable to prevent the strike from finding the bottom-right corner, with Frankfurt going ahead 4-3.
RANGERS PEN FIVE: Kemar Roofe scores, Rangers' hopes are still alive as the substitute sees his strike fly in off the post to make it 4-4.
FRANKFURT PEN FIVE: Rafael Borre scores, the Colombian rifles a brilliant spot kick into the top-left corner, before wheeling away in celebration as Frankfurt win 5-4 on penalties and are crowned Europa League winners.
MAN OF THE MATCH - CALVIN BASSEY (RANGERS)
Despite ending up on the losing side, Calvin Bassey's resilient display at centre-back was pivotal for Rangers in preventing Frankfurt's pacey attackers from running in behind and having more joy in the final third.
The 22-year-old's made a plethora of recovery tackles when the German outfit broke on the counter attack and he used his pace and strength to great effect to win back possession for the Gers.
Joe Aribo and Frankfurt pair Kevin Trapp and Rafael Borre also deserve a mention for their impressive displays in Seville, but Bassey deserves all the plaudits that come his way for his colossal efforts in the Rangers backline.
With the score at 1-1 heading into the final three minutes of extra time, substitute Kemar Roofe managed to cut the ball back from the byline into the path of Ryan Kent, and although the goal was gaping, the winger somehow failed to convert from close range, with Kevin Trapp pulling off a magnificent save to keep Frankfurt in the tie.
Kent thought it was in, Rangers supporters thought it was in, and even the Frankfurt faithful would have thought their dreams of European glory were over, but a fine stop from Trapp ensured that the final would be settled on penalties.
2 - Rangers forward Joe Aribo is just the second Nigerian to score in the final of a major European competition after Alex Iwobi (for Arsenal vs Chelsea) in the 2019 UEFA Europa League. Breakthrough. #UELfinal pic.twitter.com/91vsR7DY9t— OptaJoe (@OptaJoe) May 18, 2022
1 - This is the first time in the competition's history that consecutive UEFA Cup/Europa League finals will be decided on penalties (Villarreal beating Man Utd 11-10 on penalties last season). Shootout. #UELfinal pic.twitter.com/4XLyUyICbK— OptaJoe (@OptaJoe) May 18, 2022
1983 - Eintracht boss Oliver Glasner is the first Austrian coach to win a major European competition since Ernst Happel guided Hamburger SV to the 1983 European Cup. Victorious. #UELfinal pic.twitter.com/wkjwOKpZDv— OptaJoe (@OptaJoe) May 18, 2022
13/13 - Eintracht Frankfurt scored in all 13 of their UEFA Europa League games this season. Since the rebrand of the competition in 2009, they're just the second side to score in every single game from the group stages to the final after Chelsea in 2018-19. Euphoria. #UELfinal pic.twitter.com/G0qIrjsCSl— OptaJoe (@OptaJoe) May 18, 2022
🇩🇪 - The last time a German club won the UEFA Cup/Europa League in 1997— Gracenote Live (@GracenoteLive) May 18, 2022
▶️Frankfurt finished 7th in the 2.Bundesliga
▶️Romania (9) were ahead of England (13) in the FIFA rankings
▶️Man City finished 14th in the 2nd tier
▶️Hodgson had yet to manage his 1st PL game#UELfinal #UEL
Possession: Eintracht Frankfurt 48%-52% Rangers
Shots: Eintracht Frankfurt 22-14 Rangers
Shots on target: Eintracht Frankfurt 4-6 Rangers
Corners: Eintracht Frankfurt 11-2 Rangers
Fouls: Eintracht Frankfurt 18-10 Rangers
EINTRACHT FRANKFURT (3-4-2-1): Trapp; Toure, Ndicka (Lenz 101'), Tuta (Hasebe 58'); Knauff, Sow (Hrustic 105'), Rode (Jakic 90'), Kostic; Kamada, Lindstrom (Hauge 71'); Borre
RANGERS (4-3-3): McGregor; Tavernier, Goldson, Bassey, Barisic (Ramsey 116'); Kamara (Arfield 90'), Lundstram, Jack (Davis 74'); Kent, Aribo (Sands 101'), Wright (Sakala 74') (Roofe 116')