It was a first ever win for Celtic away to an Italian club and means they are guaranteed to progress from Group E with two games still to play.
Ciro Immobile gave Lazio an early lead but James Forrest equalised with a fine finish before half-time. Fraser Forster made some superb saves to keep the hosts at bay and Olivier Ntcham scored with a cool finish deep into stoppage time to secure a thrilling victory for the Hoops.
It was also a memorable evening for Rangers, who are well placed to qualify from Group G after beating Porto 2-0 at Ibrox thanks to second-half goals from Alfredo Morelos - his 21st of the season - and Steven Davis.
Steven Gerrard's side are level on points with leaders Young Boys, with Feyenoord - who Rangers face in Rotterdam on matchday five - and Porto three points behind.
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