In the first 90 minutes, England were twice denied by goalkeeper Yann Sommer making fingertip saves to push the ball onto the woodwork, while Callum Wilson had a late strike ruled out following a VAR check.
Sommer again came to Switzerland's rescue during the 30 additional minutes but despite a steady stream of excellent Trent Alexander-Arnold crosses England could not find the breakthrough.
Gareth Southgate's side broke their penalty malaise at last year's World Cup and, after 11 converted kicks, including one from Pickford, the Everton keeper dived to his right to keep out Drmic's poor effort and finally settle the prolonged play-off clash.
England almost opened the scoring in the second minute, but Harry Kane's clever chip bounced back off the crossbar after goalkeeper Sommer got a touch to the ball.
Kane freed Raheem Sterling for a one-on-one chance but his tame finish was blocked by Sommer, the forward then failing to convert a right-wing cross from Alexander-Arnold.
Another pinpoint Alexander-Arnold delivery created an even better opportunity for Dele Alli in the 35th minute but his header flew over the crossbar.
UEFA Nations League (@UEFAEURO) June 9, 2019
Sommer made a great save to prevent Fabian Schar scoring an own goal after the interval, turning a deflection off the defender onto the post, then at the other end Pickford was called into action to push away Granit Xhaka's firm drive.
Substitute Wilson seemed to have won it late on for England, but the strike was disallowed after a VAR check for his foul on Manuel Akanji in the build-up.
A brilliant double save by Sommer maintained the deadlock in extra time, with the woodwork saving Switzerland again when Sterling beat the goalkeeper with a 20-yard free-kick.
Sommer could do nothing about England's penalties, though, with Pickford netting himself before making the decisive save from Drmic in the shoot-out.
What does it mean? England experiment with flexibility
Southgate used a fairly irrelevant game to try a new tactical system, which saw England alternate from a traditional back four to more of a central three, with Eric Dier dropping into defence from the base of midfield.
Sharp Sommer stars for Swiss
That Switzerland stayed in the game owed much to their goalkeeper, who twice made vital interventions to turn the ball onto the woodwork. On the back of a solid season for Borussia Monchengladbach, Sommer stood out for his composure and reactions in key moments.
Tired Sterling wasteful for Three Lions
England could and perhaps should have been out of sight at half-time, but Sterling failed to make the most of two clear chances. The Manchester City winger looked like a long season had caught up with him with fatigue seemingly playing a part in an under-par display, a heavy touch in the box during the second half summing up his frustrating efforts.
Key Opta Facts
- England finished third in a major international tournament for the first time since Euro 1968.
- Switzerland have won just one of their past 23 matches against England in all competitions (W1 D6 L16).
- England scored six penalties in a single shoot-out at a competitive international tournament for the first time in their history.
- Jordan Pickford became the first goalkeeper to take (and score) a penalty for England in a competitive penalty shoot-out.
- Trent Alexander-Arnold created seven goalscoring chances versus Switzerland; the last player to create more for England in a match was James Milner versus San Marino in October 2014 (9).
Both teams will enjoy a deserved rest before the 2019-20 season. In September, England's Euro 2020 qualifying campaign sees them host Bulgaria and Kosovo, while Switzerland play the Republic of Ireland and Gibraltar.