Appearing in their first European quarter-final, and the first Danish club to reach this stage since 1997, Copenhagen continued to frustrate United as they sought a repeat of their 1-0 win when the clubs last met in the Champions League group stage in 2006.
Martial nearly won the game in normal time but again a sprawling Johnsson tipped away his curling attempt that appeared destined for the top corner.
The France striker was once more repelled by Johnsson at the start of extra time, but Martial won a penalty as United kept the attack alive and the striker was upended by Andreas Bjelland.
Fernandes blasted home from the spot on 95 minutes but Solskjaer's team were left to sweat until the end as Johnsson pulled off sensational stops from the Portuguese and substitute Juan Mata, with Victor Lindelof the third United player to strike the post.
United though ultimately ensured their 2019-20 campaign will extend into a second year, having started the season 365 days ago with a 4-0 Premier League win over Chelsea. Awaiting them on Sunday will be Sevilla - who have won the Europa League and its precursor, the UEFA Cup, a record five times - or domestic rivals Wolves.
Former United striker Romelu Lukaku grabbed the decisive goal for Antonio Conte's Inter against Leverkusen as the Serie A runners-up moved a step closer to adding to their titles from 1991, 1994 and 1998.
Inter struck first as Nicolo Barella guided in from the edge of the area on 15 minutes, with Lukaku bundling in a second six minutes later.
Kai Havertz reduced the deficit in possibly his final game for Leverkusen on 25 minutes after exchanging passes with fellow Germany international Kevin Volland, but Inter held on to seal their place against Shakhtar Donetsk or Basel in next week's semi-final.
"The defence did a tremendous job again tonight. We are growing but we have to learn to kill games off when we have the chances to do it, and we didn't tonight," said Lukaku.
"Now we have time to recover physically and prepare for another great game. We will have to be ready."