The Bombers showed tremendous ticker in their 18.9 (117) to 11.16 (82) away triumph at AAMI Stadium.
Essendon didn't score for the initial 20 minutes as they conceded the initial three goals, but then steamrolled the fancied Crows.
Midfielder Ben Howlett sparked the Bombers' comeback and capped a best-afield game with four goals while Alwyn Davey booted three.
Essendon midfielders Jobe Watson, Dyson Heppell and Brent Stanton were prominent and ruckman Tom Bellchambers shaded big Crow Sam Jacobs.
Adelaide struggled for clear winners with Richard Douglas (two goals), on-baller Matthew Wright and defender Brent Reilly among their best while Taylor Walker kicked three goals from limited chances.
Leading into the match, coach James Hird hailed his players' resilience amid the club being investigated for possibly using banned drugs last season.
And his charges demonstrated terrific character to fight back after a fast-starting Adelaide streeted to an early 22-0 advantage.
Essendon didn't kicked a goal until the 27th minute of the first quarter and appeared fortunate to only trail by 13 points at quarter-time.
But the Bombers blitzed Adelaide with six goals to one in the second quarter, with Howlett kicking two and setting up another, as the victors took a handy 14-point break into halftime.
And their lively goalsneak Davey ensured they were never headed by booting three goals in the third quarter, including one from a controversial new law.
Adelaide's Reilly slid into Davey's legs while gathering the ball, only for the Bomber to be given the free kick under the rule which penalises any sliding into an opponent's legs.
The goal helped Essendon stifle Adelaide, who repeatedly threatened a comeback but, after their initial three-goal spree, couldn't kick consecutive goals for the rest of a match attended by 42,218 spectators.
Essendon coach James Hird said the victory was based on more than just an emotional response to the club's drugs saga.
"Emotion only takes you so far," Hird said.
"I didn't find that the players before the game were overly emotional or overly excited.
"The first 10 minutes showed that maybe we kept them emotionally a bit flatter than perhaps we could have.
"But the way they played the game, it obviously means a lot to us ... because the eyes of the footballing public are on us."
Adelaide coach Brenton Sanderson said his side's glaring weakness was an inability to lay tackles on the Bombers, who landed 26 more tackles than the Crows.
"We couldn't stick tackles and they brushed us away," he said.
"A 35 point loss is not what you want in round one and is very disappointing for our footy club."