Drogba, who announced on Tuesday he was quitting Chelsea, waved at supporters as he jogged through the town centre with his torch.
Many of those who had turned out to see their footballing hero were wearing Chelsea shirts.
The Ivory Coast striker, who scored the equaliser and then the winning penalty in Saturday's dramatic Champions League final win over Bayern Munich, is leaving the club to pursue his career elsewhere.
He departs Stamford Bridge with his reputation as one of the club's greatest players firmly intact, but he knows it could have been so much different.
Ever since the 34-year-old joined the club from Marseille in 2004 they have been chasing the European Cup and, after so many near misses, Drogba's final kick for them earned them the prize they coveted.
Had he not have scored and had Bayern gone on to win, he admits he may have been tempted to stay but, with all his loose ends tied up, he is moving on to a new chapter.
Speaking to Radio 5 Live, he said: "I think winning the game was the key. Not me scoring the penalty.
"Winning the game was the turning point, that's why I have decided to say goodbye.
"We made the story together and I want people to remember that. This is the best time to move on.
"Yes, there's a chance that if the result was different, maybe I would be here.
"We were talking for many weeks, but the turning point was based on the fact I won everything I could and scored in most of the finals we have played, trying to make a difference.
"I hope everyone will understand. Everything I did was for the fans and I hope they understand."
Drogba emerged from the pedestrianised Market Street where fans had gathered to catch a glimpse of the star.
The footballer, who was the 84th torchbearer of the day, ran down Commercial Road in Swindon's bustling shopping district.
The road was turned into a sea of blue, red and white from the hundreds of Union flags, balloons and football shirts.
Drogba was flanked by police and security as he handed over the flame to Stephen Ratcliffe, chairman of Newlands Park, a community multi-sport venue owned by Cheltenham Rugby Club.
As the two torches touched, spectators ran towards Drogba to get a photo of the footballer.
He was then quickly whisked through the crowd into a silver car and driven away.
Drogba has been a talisman for Chelsea, spearheading their drive for success and delivering crucial performances in the matches which mattered most.
It been reported that Drogba's next move will take him to China to link up with former Blues team-mate Nicolas Anelka.
He would not confirm that but did say that should Chelsea ever need him once his playing days are over, he will always be happy to answer their call.
Chelsea are set for some major decisions on how to take the side forwards next season.
The Blues have yet to resolve firm plans for who will take charge of the team, with interim manager Roberto Di Matteo having steered them to FA Cup final success and the Champions League.
Chelsea will wait until Di Matteo's contract expires at the end of June before deciding on the Italian's future.