The Brazilian full-back Lodi headed in unmarked at the back post four minutes before half-time to end United's last chance of silverware this season as their drought without a trophy since 2017 goes on.
United now face an uphill task just to get back into the Champions League next season as an under-performing squad filled with star names sits fifth in the Premier League.
Atletico's own struggles to break into La Liga's top four this season have shown up their limitations.
But the experience of Diego Simeone's men in the Champions League knockouts shone through as they comfortably saw the game out to reach the last eight for the sixth time in the last nine seasons.
"We were fully aware that against this team you need to score the first goal yourself," said United interim boss Ralf Rangnick.
Cristiano Ronaldo had consistently been the scourge of Atletico in the Champions League, twice landing the ultimate blow in final defeats to Real Madrid and scoring a hat-trick for Juventus to overturn a 2-0 first leg deficit at this stage three years ago.
However, Simeone's men finally got one over on the 37-year-old as for the first time in his career he failed to register a shot in a Champions League game in which he played the 90 minutes.
Anthony Elanga came off the bench to salvage a 1-1 draw for United in the Spanish capital three weeks ago and the Swedish teenager had his side's best chance on 13 minutes.
Diogo Dalot's low cross picked out Elanga and his goalbound effort rebounded off Atletico goalkeeper Jan Oblak's head to safety.
Atletico had been penned inside their own half for the opening 15 minutes but posed a threat every time they managed to string a few passes together on the counter-attack.