Sadiku headed home the game’s only goal after 43 minutes in Lyon to leave debutants Albania facing an anxious wait to see if they qualify from Group A as one of the four best third-placed teams.
Host nation France progressed as group winners following a 0-0 draw with second-placed Switzerland as Albania finished third on three points, but Romania exited with just a single point.
“I’m sure the Albanians are really happy. We’re pleased to make people happy. This team has a big heart. They give everything, perhaps even a bit much at times,” said Albania coach Gianni De Biasi.
“The players wanted to score again and we almost had to stop them because we needed to be sensible in the final few minutes.
“Now we’re going to prepare to possibly play a game on Friday or Saturday.”
Romania coach Anghel Iordanescu made wholesale changes as Alexandru Matel replaced the injured Razvan Rat at left-back with Denis Alibec handed a start up front.
Ovidiu Hoban, Adrian Popa and Nicolae Stanciu were all recalled after the 1-1 draw with Switzerland, while Albania brought in Migjen Basha in place of the suspended Burim Kukeli in midfield.
However, there was no room in the starting line-up for skipper Lorik Cana following his return from a one-match ban having been sent off in his team’s opening game.
Stanciu fired over with an early long-range strike as Romania signalled their intentions, before a nervy Etrit Berisha smothered Hoban’s tame strike at the second attempt.
Albania were fortunate Basha stayed on the pitch when he caught Hoban with a dangerous shoulder-high lunge after just six minutes, but they had to weather the storm as Romania continued to threaten.
– Bar denies Andone –
Bogdan Stancu, scorer of both Romania’s goals in their first two games, connected sweetly with a poor headed clearance with his volley batted away unconvincingly by Berisha.
Alibec then curled narrowly wide from distance, but Albania started to impose themselves on the counter-attack as Ermir Lenjani skied over from point-blank range, although his blushes were spared by the offside flag.
Ledian Memushaj shot straight at Ciprian Tatarusanu from a narrow angle, but the Romania goalkeeper was at fault as Sadiku nodded Albania ahead just before half-time.
Memushaj’s driven right-wing cross was misjudged by Tatarusanu, who was left out of position as Sadiku headed back across goal to trigger wild celebrations among the sizeable Albanian support inside the Stade de Lyon.
Memushaj forced Tatarusanu into a low save on the hour as Albania pressed for a second, but their dreams were nearly shattered when substitute Florin Andone rattled the crossbar on 76 minutes.
The Cordoba striker raced through after a slick passing move only for his powerful strike to smack against the woodwork.
Albania, who lost 1-0 to Switzerland before going down 2-0 to France in their first two games, must wait until the completion of the final round of group matches before finding out if they qualify for the last 16.