A goal up through Joe Aribo, Gers found themselves 2-1 down at the break before levelling through a Borna Barisic penalty.
The home side hit the front again before Ryan Kent equalised again, and then a second Barisic penalty in the final minute of normal time secured another impressive away result for the Light Blues.
From the side which came through 1-0 against Kilmarnock in the Scottish Premiership on Saturday, it was a huge plus to have Scott Arfield, Kemar Roofe and Alfredo Morelos all in the starting team for manager Steven Gerrard as part of five changes overall after recent injury and suspension issues.
Indeed, Fil Helander and Glen Kamara also came back into the XI as part of five changes overall.
VAR came back into play for this round of the competition, Felix Zwayer of Germany being the official looking after that, while Bulgarian Georgi Kabakov was the man in the middle.
Gers began the game well, and had the first half chance of the match inside three minutes – Connor Goldson firing over under pressure after James Tavernier fed him from the second phase from a corner.
From that move, came the first VAR check of the match for a penalty, and while the ball came off defender Maxime Le Marchand’s arm, the officials were satisfied there was no foul committed.
Scott Arfield had a 25-yard effort as the visitors pressed well to turn the ball over in an advanced area, while Antwerp threatened for the first time from similar distance on 10 minutes, Ritchie de Laet shooting just wide of Allan McGregor’s right-hand post.
Tavernier then had a shot over, again from the 25-yard mark, while Aurelio Buta tried to curl one across McGregor from a wide position, again seeing the effort miss the target.
There was a blow for the visitors in 25 minutes, with skipper James Tavernier unable to run off a knee knock, and he had to be replaced at full-back by Leon Balogun.
Deputy captain Goldson had a deflected header land in the arms of goalkeeper Alireza Beiranvan, before the stopper denied Morelos one-on-one to the right of goal after Borna Barisic put him in.
Then, on 39 minutes, the breakthrough.
Antwerp’s high defensive line was providing opportunities for the ball to be played over the top, and that’s exactly what Goldson did for Roofe to run onto.
Goalkeeper Beiranvand came out to try and block but spilled the ball out to Morelos on the edge of the area. Beiranvand tried to recover, but Morelos managed to get it wide to the incoming Joe Aribo to fire into the empty net from 25-yards.
On the balance of the play, it was a deserved lead for the visitors, with that being Aribo’s sixth goal of the season.
However, just before half-time, Antwerp equalised. Ritchie De Laet’s free-kick from wide met the head of Felipe Avenatti to loop a header beyond McGregor and into the back of the net.
There was then two further blows for the visitors in injury-time at the end of the half, firstly, Kemar Roofe had to go off with a knock with Ryan Kent taking his place.
Then, in the seventh minute of stoppage time, Antwerp were awarded a penalty – Steven Davis was adjudged to have fouled Refaelov inside the area, with VAR confirming the referee’s decision.
Up-stepped Refaelov to take, and while McGregor guessed correctly, the precision and power on the kick was too strong and handed the visitors a half-time lead.
Gers began the second half strongly with plenty of possession, and Maxime Le Marchand cleared an Aribo effort off the line after Kamara fed him at the end of a break.
It was all Rangers, and on 58 minutes, via the VAR, they were awarded a penalty. Morelos looked to be tugged by De Laet as he tried to head home a Barisic cross, and while play raged on, referee Kabakov eventually stopped play on the advice of the VAR and went to check his monitor.
Following a short check, he pointed to the spot for Rangers first-ever VAR-awarded penalty kick. With Tavernier off the park, Borna Barisic took it, and beat the goalkeeper to his right for his third of the season.
Gers continued to dominate, but on 66 minutes, with really Antwerp’s first foray into the away half of the second half, the home side went back in front – Martin Hongla finishing from outside the box beyond McGregor.
Substitute Ianis Hagi fired wide on the end of a super Barisic delivery as Gers looked for another equaliser before that leveller did arrive on 83 minutes.
A super one-two between Kent and Morelos put Kent into the area, he danced inside the challenge of Le Marchand and fired a wonderful strike beyond sub goalie Ortwin De Wolf for 3-3.
There then followed perhaps the most unbelievable passage of play of the season.
Kent went one-on-one with De Wolf firstly, being denied by the leg of the goalkeeper. Then, at the other end, Martin Hongla was denied by an excellent McGregor stop after the Antwerp man went through.
The drama was only just beginning, however. Ryan Jack cleared off the line from a De Laet header, before Gers charged to the other end, and a Kent strike came off the hand of Abdoulaye Seck in the area.
He was shown a second yellow card as the referee pointed to the spot, leaving the Light Blues with another penalty.
Again, he opted to strike to the right, and his power and precision were excellent to incredibly put Gers 4-3 up on the night in the final minute of normal time.
Koji Miyoshi struck a post in six minutes of injury time, before finally, the referee brought matters to a close after over 100 minutes of breathless football.
The two sides meet again next Thursday at Ibrox with a place in the Last 16 of the tournament at stake.
RANGERS: McGregor, Tavernier (Balogun, 24), Goldson, Helander, Barisic, Davis, Kamara (Jack, 74), Arfield (Hagi, 74), Aribo, Roofe (Kent, 45+5), Morelos.
SUBS NOT USED: McLaughlin, Simpson, Itten, Wright, Stewart.
ROYAL ANTWERP FC: Beiranvand (De Wolf, 77), De Laet, Seck, Refaelov (Ampomah, 90+2), Boya (Versraete, 88), Gerkens (Miyoshi, 90+2), Hongla (Gelin, 90+2), Buta, Le Marchand, Avenatti, Lukaku.
SUBS NOT USED: Chevaughn, Nsimba, Pavlic, Van Den Bosch.
REFEREE: Georgi Kabakov (BUL)
VAR: Felix Zwayer (GER)