Unregistered defender Bartosz Bereszynski came on as a substitute with three minutes remaining in Legia's 2-0 win which sealed a 6-1 aggregate victory.
UEFA investigated the matter and awarded Celtic the match with a 3-0 victory, making the aggregate score 4-4 and sending the Scottish champions through on the away goals rule after they lost the first leg 4-1.
Celtic went into Friday's draw for the playoff round of matches and were pitted against Slovenian champions Maribor.
Legia, who lost reached the Champions League group stage in 1995-96, dropped into the Europa League and must make the long trip to Aktobe in Kazakhstan for the first leg of their playoff tie.
Dominik Ebebenge, sport department coordinator and head of development for the Polish club, said Legia would appeal against the decision.
"This is unjust and completely disproportionate," he told reporters at UEFA headquarters. "There have been similar cases in the past which have not punished in this way. This was just a basic human error.
"We have spent eight years working towards this and now it has been taken away from us. Financially, it's unimaginable and the players are shattered."
Bereszynski was sent off during last season's Europa League and was banned for three matches.
The fullback missed both Legia's second qualification round matches against Ireland's St Patrick's Athletic and the first leg against Celtic.
However, it was found that the club had not registered him for the games against St Patrick's, meaning the suspension had technically not been served.
"The point is that the regulations are very clear and didn't leave any space for interpretation," UEFA secretary general Gianni Infantino told reporters.
"He missed three matches but...it's clear to the clubs that in order to serve a suspension you have to be on the (players') list.
"From a footballing point of view, 6-1 is a very, very clear result, but a law is still a law and at UEFA we are always complying with the regulations, especially our own regulations.
"If you had a regulation which said something else....'may forfeit' for example....then you can start arguing, but you may end up with different discussions."
"It's never good to have a result which is not on the pitch. We are never happy," he added.
It is the second time Celtic have been reinstated to European competition after opponents were found to have used ineligible players.
In the 2011-12 season, Swiss side Sion beat the Old Firm giants in the Europa League playoffs before falling foul of eligibility rules, having used five ineligible players.
Bereszynski, 22, has been suspended for one further European match for which he would otherwise be eligible, a punishment that will be added to the remaining two-match ban.
(Additional reporting by Josh Reich, editing by Ed Osmond)
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