The Bombers trailed 37-8 early in the second quarter of Friday night's match on the back of four opening-term goals from Jake Waterman.
Things still looked grim for Essendon midway through the third term when West Coast led by 25 points following a booming 52m set shot from Elijah Hewett.
But two goals in the space of a minute from Archie Perkins helped the visitors turn the tide, with the Bombers levelling the scores by three-quarter time.
When Wright kicked his sixth goal at the 12-minute mark of the final term, Essendon led by 12 points.
In a thrilling finish, Eagles forward Jack Williams kicked a goal to reduce the margin to two points with 28 seconds remaining.
Essendon kicked the ball out wide from the next centre bounce to chew up vital seconds, with an ensuing intercept mark sealing the 11.11 (77) to 11.9 (75) win in front of 46,080 fans.
Wright was the star of the day, kicking 6.1 and gathering 17 disposals and seven marks in his first match of the AFL season.
Zach Merrett (26 disposals, seven clearances, one goal), Andrew McGrath (25 disposals) and Nic Martin (26 disposals, one goal) also played important roles.
But worryingly for Essendon, ruckman Sam Draper went down off the ball early in the final quarter with what looked to be a serious lower right leg injury.
Draper had no one next to him when he crumpled to the turf as he set off to run.
Vision later showed a devastated Draper being consoled in Essendon's change room while his right leg was elevated on a chair.
Bombers wingman Xavier Duursma was injured in the dying minutes when he sustained an accidental knee to the midriff from Eagles ruckman Matt Flynn.
For West Coast, Waterman finished with four goals and 10 marks, while Jeremy McGovern (23 disposals) was a rock in defence.
Tim Kelly collected 22 disposals and three clearances in his first match back from being dropped to the WAFL, and Elijah Hewett (22 disposals, five clearances, two goals) played his best match of the season.
Harley Reid was restricted to 17 disposals and three clearances for the match, and his heavy first-quarter bump on Archie Roberts could come under scrutiny.
West Coast tallied an AFL record-low 77 contested possessions against Carlton last week, but they produced 35 alone in the opening quarter against Essendon as Waterman took the game by the scruff of the neck.
The scoreline read 30-8 by the time Waterman kicked his fourth goal after the quarter-time siren, marking just the second time West Coast had won a term this season.
Alarm bells were ringing for Essendon when Jamie Cripps kicked a goal early in the second quarter to stretch the margin to 29 points.
But two quick goals to Wright - on the back of Essendon's 10-4 clearance dominance for the term - got Essendon back into the match, reducing the margin to 18 points by half-time.
When West Coast youngster Hewett kicked a booming 52m set shot, the Eagles led 61-36 at the 18-minute mark of the third quarter.
Perkins kicked a set shot and then a snap in the space of the minute to haul Essendon back in the match.
And when Draper nailed his set shot and then Wright snapped his fifth, scores were level with a quarter to go.
Martin flew high for a mark and kicked truly to kick off the final quarter, and Wright followed it up with his sixth goal of the match to give Essendon a decisive advantage.