Michel's team fell behind to an early strike from Asteras' Spanish captain Rayo, but the Reds drew level just before half-time thanks to David Fuster's goal.
Substitute Rafik Djebbour rammed home from a corner in the first period of extra time to make it 2-1.
Djamel Abdoun then added gloss to the scoreline with a penalty in the 119th minute after he was adjudged to have been fouled by Sebastian Bartolini.
The match was held up for several minutes after Olympiakos' third goal when tempers boiled over and several skirmishes broke out, resulting in red cards being shown to Olympiakos' unused substitute Francois Modesto and Tripolis striker Emmanuel Perrone.
The victory sealed a 16th double for Olympiakos, who also wrapped up their 40th league title in March with five games to spare.
Despite Olympiakos being the overwhelming favourites, Tripolis enjoyed the better of a tense encounter, with a long-range effort from Abdoun Olympiakos' only effort in the opening exchanges.
Goalkeeper Roy Carroll then produced a reflex save from Pablo De Blasis's header in the 10th minute
Tripolis took the lead four minutes later. A neat move down the left released Ximo Navarro on goal, and although his effort was saved by Carroll, Rayo followed up to sidefoot home the rebound.
Olympiakos responded well and drew level with a fine move of their own down the left, Abdoun picking out the run of David Fuster with a dinked cross that the Spaniard slid home beyond Tripolis keeper Marton Fulop.
Tripolis were denied what looked like a clear penalty in the 67th minute when Giannis Maniatis handled Khalifa Sankare's header off the line.
Olympiakos raised the tempo early in extra time and were rewarded seven minutes in when Djebbour powered in the rebound after Fuster's header was saved following an Ariel Ibagaza corner.
Tripolis were denied another penalty in the closing minutes when Rayo looked to have been hauled down by Dimitris Siovas, before their miserable evening was completed when Olympiakos were awarded a spot-kick by referee Athanasios Giachos for a foul that replays suggested seemed to have been committed outside the box.
Abdoun converted the spot-kick and sealed a barely deserved triumph for Olympiakos.
- Sports & Recreation
- David Fuster
- Djamel Abdoun
- Rafik Djebbour