As many as seven feats were smashed at the Allianz Arena in Munich, where Germany recorded the biggest-ever opening victory in the history of the men's European Championships.
Florian Wirtz began the goal rush by becoming the youngest Germany scorer in Euros history, before fellow 21-year-old Jamal Musiala also got in on the act with a thunderous finish into the roof of the net.
Kai Havertz then coolly slotted home the hosts' third from the penalty spot after Ryan Porteous's sickening tackle on Gundogan - which the Watford man was sent off for - before Fullkrug and Emre Can came off the bench to fire in long-range efforts in the second period.
Antonio Rudiger's own goal did prolong Germany's streak without a clean sheet in major tournaments to 13 matches, but there was no quelling the celebratory atmosphere in Munich, while the Tartan Army's Euros party quickly came to a halt.
Gundogan highlights Scotland "warning" despite emphatic win
Gundogan - who was fortunate to escape a serious injury after falling victim to Porteous's studs-up tackle - labelled Germany's display "simply fantastic" but admitted that Scotland's consolation goal was a warning sign for future contests.
"We played with real intensity from the first minute and took risks. Our first-half performance was simply fantastic, and it was just how we wanted to start," the DFB website quotes the Barcelona midfielder as saying.
"We need this kind of atmosphere and support to go deep in the tournament. Their goal, however, showed that if you switch off even for a minute, you can be punished for it. It was a good warning for us."
Head coach Nagelsmann, meanwhile, drew particular attention to the Germans' "power" in the first 20 minutes as they overwhelmed their visitors, saying: "The first 20 minutes were impressive.
"We played with lots of power and pressed them, which really helped us. We got early goals and were focused until the final whistle. It's great that the goals were spread out across several players. Tonight was a good performance, but only the first step."
Niclas Fullkrug: 'We never let up'
Finally, Borussia Dortmund striker Fullkrug - who lost out to Havertz for a starting spot but was effective off the bench - applauded his side for always striving for more goals no matter what the scoreline read.
"We took a few minutes to get into the game at the start, but after that we improved and kept gaining confidence. We never let up after any goal we scored. We're pleased with this fantastic start," the striker added.
In contrast, Scotland skipper Andrew Robertson admitted that his side were outclassed in all areas of the pitch, although his post-game comments were blasted as "rubbish" by former Manchester United captain Roy Keane.
Already sitting pretty at the top of the section, Germany now gear up to meet Hungary at the MHPArea in Stuttgart next Wednesday, a few hours before Scotland endeavour to bounce back against Switzerland in Cologne.