Martin launched from 55 metres on the run for the final goal as the Bombers prevailed 10.18 (78) to 11.9 (75) at Adelaide Oval on Friday night.
Essendon climb into the top eight with a fourth win of the season but the Crows remain in the mire with one win and five losses.
Martin's strike with about eight minutes left put the visitors six points up and the Bombers then held on bravely despite a series of Adelaide attacks.
"It ebbed and flowed and could have gone either way at the end, obviously," Essendon coach Brad Scott said. But I am pretty comfortable saying that our guys deserved the end result."
Essendon skipper Zach Merrett was a standout with 28 disposals, seven clearances and a crucial last-term goal while Jade Gresham (26 touches) and Jye Caldwell (26) were also busy.
Martin capped his influential 27-disposal display with the late game-breaker while Harrison Jones and Kyle Langford kicked two goals each.
Zach Merrett was inspirational for the Bombers, with a team-high 28 disposals. (Matt Turner/AAP PHOTOS)
Adelaide's Jake Soligo continued his break-out form in the midfield with 31 disposals, 10 clearances and a goal.
His captain Jordan Dawson was superb with 31 touches, two goals and nine tackles while Mitch Hinge (25 possessions) and Josh Worrell shone in defence.
"We looked slow, reactive early which put us on the back foot," Crows coach Matthew Nicks said. "The game wasn't on our terms at all ... a disappointing night."
The Bombers started brightly and held a slight edge at quarter-time, 2.4 to 2.2. Essendon's early efforts were largely built by skipper Merrett, who dominated the opening term with 13 disposals and four clearances.
The visitors then cashed in with three consecutive goals, and when Jake Stringer snapped accurately they held a 22 point lead.
Jordan Dawson was in everything for the Crows, including this tussle with Sam Durham. (Matt Turner/AAP PHOTOS)
But Adelaide clawed their way back in to the contest with captain Dawson scoring two of their three unanswered goals.
Crow Darcy Fogarty had a shot after the halftime siren to level scores but missed with Essendon, holding a 31-16 inside 50 advantage, up 5.9 to 5.4 at the long break.
The third quarter swung wildly, with Fogarty triggering a four-goal burst in nine minutes. Adelaide's scoring spurt created a 19-point lead but just when the Bombers appeared down and almost out, they rallied.
The ever-lively Jones snared his second goal and his dangerous teammate Langford then scored two late majors - his set-shot on the siren gave the Bombers a three-point buffer.
Essendon crept five points up until the 10th minute of the last quarter when Adelaide's Ben Keays goaled after a free kick to give the home side a one-point lead.
Crow livewire Izak Rankine soon produced a snap from heavy traffic which bounced through for a seven-point break with 12 minutes left.
The Bombers' inspirational Merrett answered his team's predicament, swooping on a loose ball and converting to again reduce the margin before Martin's long-range conversion gave the visitors enough breathing space.
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