Carrick netted three late goals to complete an amazing comeback in the relegation battle at Seaview.
Gary Hamilton fired Glenavon ahead early in the second half and Mark Miskimmin struck twice to make it 3-0.
Shane Dolan missed a penalty but Anto Lagan netted from the spot before Brian McCaul scored for the visitors.
The late drama started with Marcus Kane's goal on 85 minutes and Thomas Hyland made it 4-3 before Paul Heatley equalised late into injury-time.
It is a crucial point gained for Rangers, who remain bottom but are now just three points adrift of Dungannon Swifts and four behind Lisburn Distillery and Glenavon.
Carrick edged a first half of few chances and all eight goals came in a thrilling second half.
Glenavon player/boss Hamilton hit the bottom corner after 47 minutes and a double from Miskimmin quickly followed.
The former Linfield striker headed in from a Hamilton cross before cutting inside and firing low into the Carrick net.
Kyle Neill's handball on 66 minutes resulted in a penalty but keeper Andy Coleman pushed Dolan's effort onto the crossbar.
Neill was again guilty of conceding a spot-kick, this time for a foul on Heatley, and Lagan converted with 11 minutes left.
McCaul stabbed in to restore the three-goal advantage but Kane shot past Coleman to start the comeback and Hyland headed home in the last minute of normal time.
Heatley was the last-gasp hero, the winger lobbing the ball over Coleman to equalise in the fourth minute of added time.