The 36-year-old, who rejoined the club in the summer after nine years away, made a telling debut with a 90th-minute winner to get the Shrimps off and running for the new season.
Jim Bentley's side had to come from behind after debutant Mohamed Eisa gave the Robins the lead two minutes from the break in controversial circumstances.
Kyle Storer swung a free-kick in from from the left and after Danny Wright headed across goal, the summer signing was on hand to poke the loose ball home from close range - with the Shrimps furious that a push by Wright had not been seen by referee Antony Coggins.
The visitors went close to scoring a second six minutes after the restart when Jordan Grimes saw a close-range volley hit the post after an Alex Kenyon clearance fell perfectly for him.
Three minutes later the Shrimps levelled through Thompson, who slid in at the far post to steer home an excellent low cross from Aaron Wildig after a swift Morecambe counter-attack.
Thompson then provided the telling moment of a tight game at the death when he volleyed home from Vadaine Oliver's header across goal to take the points.
Match report supplied by the Press Association.
Match ends, Morecambe 2, Cheltenham Town 1.
Second Half ends, Morecambe 2, Cheltenham Town 1.
Foul by Michael Rose (Morecambe).
Jordan Cranston (Cheltenham Town) wins a free kick on the right wing.
Attempt saved. Dan Holman (Cheltenham Town) right footed shot from the left side of the box is saved in the centre of the goal.
Substitution, Morecambe. Sam Lavelle replaces Garry Thompson.
Substitution, Cheltenham Town. Dan Holman replaces Nigel Atangana.
Goal!  Morecambe 2, Cheltenham Town 1. Garry Thompson (Morecambe) right footed shot from the left side of the six yard box to the top left corner. Assisted by Vadaine Oliver.
Vadaine Oliver (Morecambe) wins a free kick in the defensive half.
Foul by William Boyle (Cheltenham Town).
Attempt missed. William Boyle (Cheltenham Town) right footed shot from the right side of the six yard box is close, but misses to the right.
Foul by Adam Campbell (Morecambe).
William Boyle (Cheltenham Town) wins a free kick in the attacking half.
Attempt missed. Carl Winchester (Cheltenham Town) right footed shot from outside the box is high and wide to the left.
Attempt blocked. Carl Winchester (Cheltenham Town) right footed shot from outside the box is blocked.
Substitution, Morecambe. Adam Campbell replaces Aaron Wildig.
Aaron Wildig (Morecambe) is shown the yellow card for a bad foul.
Foul by Aaron Wildig (Morecambe).
Jerell Sellars (Cheltenham Town) wins a free kick in the defensive half.
Attempt missed. Michael Rose (Morecambe) left footed shot from outside the box is close, but misses to the left.
Foul by Michael Rose (Morecambe).
Nigel Atangana (Cheltenham Town) wins a free kick on the right wing.
Foul by Mitchell Lund (Morecambe).
Jerell Sellars (Cheltenham Town) wins a free kick in the defensive half.
Corner,  Morecambe. Conceded by Jamie Grimes.
Foul by Rhys Turner (Morecambe).
Carl Winchester (Cheltenham Town) wins a free kick on the left wing.
Substitution, Morecambe. Rhys Turner replaces Kevin Ellison.
Attempt missed. Kyle Storer (Cheltenham Town) right footed shot from outside the box misses to the right.
Jerell Sellars (Cheltenham Town) wins a free kick in the attacking half.
Foul by Michael Rose (Morecambe).
Attempt missed. Michael Rose (Morecambe) left footed shot from outside the box is just a bit too high from a direct free kick.
Vadaine Oliver (Morecambe) wins a free kick in the defensive half.
Foul by Jordan Forster (Cheltenham Town).
Foul by Dean Winnard (Morecambe).
Mohamed Eisa (Cheltenham Town) wins a free kick in the defensive half.
Corner,  Cheltenham Town. Conceded by Alex Kenyon.
Corner,  Cheltenham Town. Conceded by Barry Roche.
Attempt saved. Kyle Storer (Cheltenham Town) right footed shot from outside the box is saved in the bottom right corner.
Foul by Michael Rose (Morecambe).
Two goals from the returning Garry Thompson gave Morecambe a dramatic opening day win over Cheltenham.