Southgate's preparations for the match were thrown into flux when Phil Foden and Mason Greenwood were banished from the squad for breaking coronavirus protocols in the aftermath of Saturday's 1-0 win over Iceland.
Conor Coady and Kalvin Phillips made their debuts but a solitary blocked shot was all England had to show for their forgettable first-half efforts, with Kasper Dolberg almost giving Denmark a deserved lead.
Harry Kane had a stoppage-time effort hacked off the line by Mathias Jorgensen when Kasper Schmeichel erred, but that near-outcome would have flattered England.
Southgate's tactical switch saw his side control significant chunks of possession early on, although the front three of Raheem Sterling, Jadon Sancho and captain Kane were frequently isolated.
Denmark went closest to the breakthrough before half-time - Barcelona forward Martin Braithwaite flashing a 31st-minute shot past the post from the edge of the area before Dolberg brought a fine save from Jordan Pickford, Christian Eriksen having left former Tottenham team-mate Eric Dier in his wake.
Eriksen clumping a 65th-minute free-kick high into the stand behind Pickford was a less astute contribution to a contest that continued to plod.
England belatedly crafted a clear opening with a little over 20 minutes remaining, with Kane unable to convert Kieran Trippier's left-wing cross at the far post.
Kasper Schmeichel then saved unconvincingly from Sterling, with Mason Mount's introduction in place of Sancho having belatedly injected some playmaking impetus for the visitors.
Eriksen hacked over from 10 yards under timely pressure from Joe Gomez before Kane almost made it two last-gasp wins from two.
What does it mean: Southgate must ditch imbalanced plan B
Fingers will be pointed at England's formation after this tepid affair, but Southgate's squad is full of players used to switching systems frequently under some of the most innovative coaches in the game. The problem here was a lack of balance, with right-footed players deployed in the left of the back three, on the left flank and on that side of the forward trident.
Mount's introduction paid obvious dividends and should have come sooner, although Southgate does have the inconvenience of planning around Foden and Greenwood's indiscretions as mitigation when assessing a bad night at the office.
Eriksen shows his class
The experienced playmaker has a point to prove at Inter this season but frequently posed eloquent questions that seemed beyond England's midfield. His three shots were more than any other Denmark player.
Rice turns in stodgy showing
Southgate is committed to giving Rice a run of games as his midfield anchorman and it is time for the West Ham man to start repaying some of that faith. His pass completion rate was only 76.7 per cent, with debutant Phillips looking far more adept in a game both players did not need to start.
Matchday three in Group A2 sees England host Belgium, with Denmark travelling to Iceland. Roberto Martinez's men have six points after dismantling Iceland 5-1.