Lennon's agent, Martin Reilly, told BBC Scotland that Lennon had coins thrown at him and was spat upon during the game at Tynecastle.
It is understood Lennon walked out of the stadium shortly after Adam Rooney scored the third goal in the 62nd minute of Aberdeen's 4-0 victory.
Lennon was sitting in the main stand at Tynecastle, yards above the technical area where he was attacked by a Hearts fan during a game in 2011. The stand was housing Aberdeen fans on Saturday.
Celtic hit out at the behaviour of some supporters in the stand.
A club spokesperson told Press Association Sport: "Clearly it is totally unacceptable for the Celtic manager to be treated in this way.
"Neil has complimented the way in which the situation was handled by stewarding staff at Tynecastle.
"We have an excellent relationship with Aberdeen Football Club and we are sure they will look into this matter."
Aberdeen, who reached their first cup final since 2000, pleaded for supporters to report those responsible.
In a statement on their official website, a Dons spokesperson said: "The support we enjoyed at Tynecastle was absolutely fantastic and helped to create an incredible atmosphere that the manager and players acknowledged played a major part in what was wonderful day and occasion for Aberdeen Football Club.
"However, it would appear that a tiny minority within the main stand behaved in a manner that is completely unacceptable and has absolutely no place in football and we would urge anyone who has any information as to the identity of those responsible to get in touch with the club in order that appropriate action can be taken."
Quoted on the BBC Scotland.website, Lennon's agent, Reilly, said: "I'm absolutely raging about the treatment of Neil, about the stewarding and the fans' behaviour. It's scandalous that Neil is treated like this."
It is understood Lennon watched the match in peace during the first half but then sat in a different seat after half-time. The Celtic boss was watching Aberdeen ahead of the two teams' William Hill Scottish Cup fifth-round meeting at Parkhead next Saturday.
Police Scotland said they had received no reports surrounding the incident.
A spokesperson said: "Police in Edinburgh have received no reports relating to assaults and abuse of anyone associated with Celtic Football Club or Heart of Midlothian Football Club following the match at Tynecastle today."
A Hearts supporter was jailed for eight months for breach of the peace after running on to the track and throwing himself at Lennon during a game in May 2011.
Two teenagers briefly held up play during Saturday's semi-final when they ran on to the pitch from the St Johnstone end and got to the technical area before being apprehended by police and stewards.
Lennon, 42, has been the victim of a number of other crimes during his spell as player, coach and manager of Celtic.
Two students were fined after Lennon was attacked in his car in Glasgow in May 2003. A year later, threatening graffiti was daubed outside his home and he was the victim of road rage when he was repeatedly yelled at on the M8 motorway.
In September 2008, he was treated in hospital after being attacked hours after an Old Firm game. Two Rangers fans were later jailed for two years each.
A package of bullets addressed to Lennon was intercepted at a sorting office in Country Antrim, in January 2011 and two men were jailed for five years in April 2012 for sending Lennon parcel bombs.
- Sports & Recreation
- Aberdeen Football Club
- Celtic Football Club