After suffering a heavy 5-1 defeat to Germany, Scotland received a vital confidence boost in the 13th minute when Scott McTominay fired his country into an early lead with the help of a wicked deflection.
They looked relatively comfortable until Xherdan Shaqiri took full advantage of Anthony Ralston's loose pass to curl an unstoppable strike out of the reach of Angus Gunn.
For a brief moment, Dan Ndoye thought he had put Switzerland in front, before turning to see the assistant's raised flag, ensuring the two nations went into the break on level terms.
The second half did not yield a goal, but it certainly produced plenty of drama as Grant Hanley rattled the frame of the goal, before Breel Embolo had a goal ruled out for offside.
Scotland ultimately had to settle for a point, meaning they will almost certainly need to beat Hungary on Sunday to progress to the knockout rounds of a major tournament for the first time in their history.
Shaqiri stunner cancels out McTominay's opener
Scotland boss Steve Clarke made two changes for the crucial encounter - one enforced following Ryan Porteous's red card against Germany.
Hanley came in for the suspended Watford man, while Brighton & Hove Albion's Billy Gilmour was brought in for Ryan Christie.
In the opposite dugout, Murat Yakin opted to make just one alteration, replacing one of Saturday's goalscorers, Kwadwo Duah, with the experienced Xherdan Shaqiri.
Having put Friday's disappointment behind them, the Tartan Army delivered a spine-tingling rendition of Flower of Scotland, giving their players a much-needed boost ahead of kick-off in Koln.
The vocal Scotland contingent had to wait just 12 minutes to celebrate their sixth goal in European Championship history.
Manchester United's McTominay latched onto Callum McGregor's neat layoff to drill a left-foot strike towards goal, before it deflected off Fabian Schar and past an unmoved Yann Sommer.
Scotland looked relatively comfortable until Shaqiri produced a piece of magic to pull Switzerland level and punish Anthony Ralston for a costly mistake.
The former Liverpool man pounced on Ralston's loose backpass to bend a powerful long-range strike into the top corner to celebrate his 32nd international goal.
The stunning shot swung the momentum firmly in Switzerland's favour, forcing Scotland to survive a period of intense pressure.
Gunn made a strong save to keep Ndoye at bay just moments before the Switzerland forward beat the goalkeeper to the ball to slot home, only for the effort to be chalked off for an offside.
Scotland have to settle for point
Ndoye continued to offer a threat after the interval, outmuscling Kieran Tierney - who picked up an injury in his bid to win to stop the striker - before dragging his low shot just inches wide of Gunn's right-hand post.
It was then Scotland's turn to go agonisingly close as Hanley used his strength to get ahead of Schar and meet Andy Robertson's whipped free-kick, sending his header crashing against the upright.
Not long after, Scotland breathed a huge sigh of relief when the assistant raised his flag to deny Embolo, who had raced through on goal and calmly lifted the ball over the onrushing Gunn.
Burnley Zeki Amdouni then spurned a golden opportunity to win the game in the 88th minute when he sent his header wide from Fabian Rieder's deep free-kick.
A draw ultimately proved to be the final result, ensuring Scotland's 28-year wait for a major tournament win will continue until Sunday at the very least.
Man of the Match - Scott McTominay
Shaqiri could be credited with the award just for producing a spectacular effort to add to the many wonder goals we have seen at Euro 2024.
Liverpool's Robertson should also get a mention after putting in a captain's performance to create numerous chances as well as playing a key role in his side's defensive efforts.
However, McTominay gets the nod for his impressive display which included his ninth international goal, and the strike which helped his country clinch a crucial point.
Not only did he provide a goal threat, but the 27-year-old also showed great energy to get through an enormous amount of work in all areas of the pitch in Koln.
Scotland's last-16 hopes to be decided on Sunday
Scotland's most recent major tournament win took place against Switzerland at Euro 1996, but they failed to repeat history at Euro 2024, leaving them in third place in Group A with one point to their name.
They know they need to beat Hungary to have any chance of clinching a best third-place spot or even second position in the group.
As for Switzerland, they are on the brink of securing a third consecutive last-16 appearance at the European Championships.
However, they still have to negotiate a tricky fixture against tournament hosts Germany, who have already sealed their last-16 spot following back-to-back victories.