The Demons trailed by 12 points early in the third quarter of Saturday night's match, but they kicked nine of the next 10 goals to secure the 16.12 (108) to 11.10 (76) in front of 41,991 fans.
Melbourne's third win on the trot improved their record to 3-5, while West Coast (0-8) remain winless and on bottom of the table.
Gawn was the pivotal figure, racking up a career-high 35 disposals to go with 47 hitouts, nine clearances, 12 score involvements and a goal.
The 33-year-old also played match-winning roles in recent wins over Richmond and Fremantle, and he helped Melbourne win the clearance battle 48-26 and inside 50m count 59-43 against the Eagles.
Demons forward Jake Melksham could be in trouble for his body shove on Jeremy McGovern that catapulted the star Eagles defender into a marking contest.
McGovern slammed into the back of Harrison Petty after receiving the shove, leaving him stunned and feeling his jaw.
The premiership defender was subbed out at halftime, and his absence was heavily felt by West Coast as the Dees dominated in the second half.
Melbourne also subbed out Petty at halftime due to concussion, possibly from the same McGovern incident.
West Coast debutant Bo Allan also faces a nervous wait for his swinging arm to the head of Jack Viney as the Demons tagger took a mark.
Christian Petracca finished with 29 disposals, eight clearances and a goal for the Demons in the absence of Clayton Oliver (personal reasons), while tagger Viney (18 disposals, seven clearances) restricted Harley Reid to just 14 disposals and no clearances.
Eagles midfielder Tim Kelly finished with just 14 disposals and three clearances.
West Coast co-captain Oscar Allen did some nice things in an under-siege defence, while Jake Waterman and Elijah Hewett kicked three goals apiece.
The opening quarter was a horror show of errors as both sides butchered the ball at critical moments.
Melbourne were at least a force at the clearances - winning the first 10 of the match - with goals to Petracca and Bayley Fritsch giving them a 14-5 lead at quarter-time.
The Demons continued to dominate the clearances in the second term but their wayward return of 1.5 came back to haunt them.
West Coast didn't score their first goal of the match until Jack Williams' set-shot shank floated through at the 17-minute mark.
It sparked a run of four consecutive goals - including a spectacular banana from the pocket from Jayden Hunt - as the Eagles finally brought the crowd to life.
Demons coach Simon Goodwin was understandably frustrated that his team didn't have the lead at halftime.
They would have grown after watching Jamie Cripps kick the opening goal of the third quarter.
But with McGovern no longer there to fend off Melbourne's forays forward, the Demons finally cashed in on their clearance dominance, slamming through six goals for the term to take a 29-point lead into the final change.
Gawn, who had 0.7 to his name for the season up until that point, put the icing on the cake with a set shot goal in the final quarter.
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