Lieut Maxse, who was serving on Lord Raglan's staff, wrote that Captain Louis Nolan, a 36-year-old officer, was responsible.
His letter was discovered with other documents inside the British Library, reports Patrick Sawer at the Sunday Telegraph.
Capt Nolan was the messenger who passed on Lord Raglan's orders to Lord Lucan - the officer who carried out his instructions.
He instructed him to "follow the enemy and try to prevent the enemy from carrying away the guns".
Professor Saul David, a military history teacher at the University of Buckingham, has studied Maxse's letter and said Capt Nolan exaggerated Raglan's orders.
Capt Nolan told Lord Lucan: "Lord Raglan's orders are that the cavalry should attack immediately: 'There, my Lord! There is your enemy! There are your guns!'"
But Capt Nolan was one of the first men to die after being struck by shrapnel from an exploding shell.
In his letter Lieut Maxse wrote: "On looking to the left, saw poor Nolan lying dead who 10 minutes before I had seen eager & full of life, galloping down to Lord Lucan, anxious & determined to make him do something with the cavalry (of which he is a member)."
He added that Nolan resented the behaviour of the men.
Lieut Maxse wrote: "He was always very indignant at the little they had done in this campaign & bitter against Lord L.
"All the cavalry lay this disastrous charge on his soldiers & say that he left no option to Lord L to whom they say his tone was almost taunting on delivering the message - if he was to blame he has paid the penalty."
Nigel Kingscoteé, a second staff officer, also said that Capt Nolan was to blame and told Raglan's son he would have "been broke by court martial" if he survived.
Prof David wrote in the BBC History Magazine that "Nolan bears the chief responsibility for what followed".
He said: "So contemptuous was he of Lucan's ability, so desperate for the cavalry to show its worth, that he failed in the one essential task of a staff galloper: to provide the officer in receipt of the message with the necessary clarification.
"It seems, moreover, that he used the word 'attack' when Raglan had intended a mere show of force. If so, Nolan bears the chief responsibility for what followed."