After picking up just one point from their first four Group E matches, the Hoops entered the contest at the Stadio Olimpico needing to beat Lazio to preserve any hope of securing a top-two finish or even claiming third spot to drop down into the Europa League.
Celtic showed their resilience for large parts of the match and had a few half-chances to open the scoring in the second half through Kyogo Furuhashi and Yang Hyun-Jun.
However, Lazio legend Ciro Immobile came off the substitutes' bench to score twice in closing stages, firstly firing a deflected strike beyond Joe Hart in the 82nd minute before coolly rolling in a second three minutes later, to help the hosts book their place in the last 16 and send Celtic packing from Europe altogether.
Celtic, who sit nine points adrift of second-placed Lazio in Group E, have been eliminated from the Champions League group stage for the second successive season, this time with a game to spare, and their wait to reach the last 16 for the first time since 2012-13 goes on.
Defeat in Italy for Celtic has also extended their winless run in the Champions League to 15 matches, becoming the first-ever British club to put together a run this long without a win in the competition's history.
In addition, Celtic have conceded at least once in each of their last 26 Champions League games against teams from Europe's top-five leagues, dating back to December 2008 when they beat Villarreal by a 2-0 scoreline.
Discussing Celtic's disappointing Champions League exit, manager Brendan Rodgers has encouraged the club's hierarchy to "add quality" to his squad in the upcoming transfer windows if they wish to compete at the highest level.
"It's the overall quality," Rodgers told reporters as quoted by BBC Sport. "We need to have our very best players available.
"What makes the difference is that bit of experience and genuine quality. That's something we can hopefully resolve over the next couple of windows.
"This group has gained more experience and shown they can compete in some games - but we need to add quality. That's the glaring thing that stands out."
Rodgers believes his Celtic side played "too safe" in the first half, saying: "We kept the ball but we needed to progress the game that bit more.
"Second half, we were much better, progressing the game through the pitch and managing the game well and then we arrive on the ascendency. In the last 20 minutes we looked like the team that could go on and get the result, but we gave away two poor goals.
"The first one, we're in a great position up the pitch and we lose possession and never recover our shape. Then they get a bit of luck with the deflection. That was a tough one to take. For the second goal, we obviously need to be stronger and better."
"We want to finish (the group) on a high," Rodgers added. "We want to win in front of our own supporters.
"Our support has been incredible, travelling all round Europe. They haven't seen us win and we'd love to give them that victory at home and finish off with a positive result."
Celtic, who have only won one of their last four games in all competitions, will endeavour to return to winning ways when they travel to Perth to face St Johnstone in the Scottish Premiership on Sunday.