Melksham fell and landed on his wrist during an intraclub match at Tullamarine on Friday. Scans revealed a small break which will keep him out of action for two-to-three weeks.
The 21-year-old will miss the opening round of Essendon's NAB Cup campaign next week, but could be available for the Bombers' clash with Richmond in Wangaratta on March 2.
"Jake will have the operation today but he will be back into training on Monday," coach James Hird told essendonfc.com.au.
"We look forward to him playing in the next two to three weeks."
In January, Essendon captain Jobe Watson highlighted Melksham as a standout in the club's pre-season so far.
Melksham was one of only two Bombers (alongside Watson) to play every game in the Bombers' injury-riddled season in 2012.
Watson will also miss the Bombers' first week of NAB Cup with a knee ligament strain.
The injuries end a shocking week for the Bombers, which on Tuesday revealed it is being investigated by the Australian Sports Anti-Doping Authority and the AFL for supplement use at the club.
Jake Melksham is a $324,600 midfielder in NAB AFL Fantasy. He averaged 63 points in 2012. Register your team at our AFL Fantasy Hub.
?Follow AFL website reporter Callum Twomey on Twitter at @AFL_CalTwomey.