Southgate's reign appeared to be ending on a rotten note when Ivan Schranz deservedly propelled Slovakia into a first-half lead, and subsequent attempts from England to claw their way back into the contest were pitiful and pedestrian.
A disallowed Phil Foden goal for offside appeared to compound England's misery, but with one of the last attacks of normal time, Jude Bellingham produced a moment of magic with an outrageous bicycle kick to force an additional 30.
The rejuvenated Three Lions required just 54 seconds of the extra-time period to turn the tide through a Harry Kane header, and even though England reverted to type in the closing stages - sitting back and inviting pressure - their efforts were just about enough.
A meeting with Italy's conquerors Switzerland now awaits the Euro 2020 runners-up, who had many a lucky star to thank for their progression, which led to sighs of relief up and down the land.
Bellingham brilliance and Kane heroics rescues embarrassing England
For the hour or so that followed after Schranz poked Slovakia ahead, it appeared that nothing England could throw at the Falcons' wall would stick, as Francesco Calzona's men defended for their lives and then some.
None of England's attackers were doing themselves justice and looked on course to be exiting the tournament with a whimper, that was until a Kyle Walker long throw in added time fell onto the head of Marc Guehi, who flicked it on for Bellingham.
The Real Madrid man had been tipped by some to enjoy a rest on Sunday on account of his tired performances in the draws with Denmark and Slovenia, but as Bellingham does, he pulled the rabbit out of a hat with an outrageous bicycle kick to save England's bacon.
Barely 70 seconds had passed in added time before a recycled free kick sparked more English pandemonium, as the two big men up top - Ivan Toney and Kane - linked up aerially for the latter to nod home and complete an unforeseen turnaround.
Silky Schranz capitalises on England's lackadaisical defending
The opening exchanges of the Gelsenkirchen tie were summed up by Slovakia's wingers running England's full-backs ragged, in particular Lukas Haraslin doing a number on Kyle Walker out on the right-hand side, and an attack on the other flank led to Slovakia's opener.
Neither Guehi nor John Stones covered themselves in glory when 37-year-old Juraj Kucka won a header to find David Strelec, who fronted up to Stones and waited patiently for Schranz to make the run into the box.
A slightly stricken Guehi - who had hurt his shoulder in the aforementioned aerial challenge - could not keep up with Schranz nor touch him, having already been cautioned, while Kieran Trippier was nowhere to be seen either.
Despite being slightly off balance, Schranz reacted to Jordan Pickford rushing off his line with a toe poke into the far corner, thus drawing level with Jamal Musiala and Georges Mikautadze on a tournament-high three goals, although he will not have the chance to add to that tally.
England are through... but they do not deserve to be
Had a boy from Birmingham not become a world-beating midfielder for club and country, England would currently be clearing out their German lockers and preparing for a miserable flight back to Blighty, and Southgate will be all too aware of the fact.
None of the Three Lions' group-stage performances provided any major cause for optimism, and nor did Sunday's last-16 tie, for all the pre-game talk of thing being different now that England were back in knockout action.
Fans may have let loose and belted out Sweet Caroline at full volume following 120 gripping minutes, but once the dust settles, Southgate and co have some serious problems to solve, otherwise a plucky Switzerland side will be the next to prevent football from coming home.
What next for England and Slovakia?
A five-day break - and a much-needed one at that - awaits the Three Lions before they head to the Merkur Spiel-Arena in Dusseldorf, where Switzerland will be their Euro 2024 quarter-final opponents.
That encounter will mark Southgate's 100th match in charge of England, while Slovakia are left to rue a second last-16 exit from three tournaments, but Calzona remains under contract until December 2025 and will seemingly be leading the Falcons into their World Cup 2026 qualifying campaign.