Cheltenham skipper Marlon Pack headed the 56th-minute winner, his sixth goal of the season, a minute after Plymouth substitute Juvhel Tsoumou had cancelled out Steve MacLean's deflected opener from a 23rd-minute free-kick.
Fletcher, whose side had flirted with relegation afterthe club came out of administration in October, said: "Both on and off the pitch it was about the bigger picture.
"As the season was going on we probably thought we might have gone down to the last game or the last couple of games, but thankfully for us it didn't.
"As a whole, looking at the bigger picture of the club and as a football team, from where we were to where we have come has been a massive, massive step.
"There is a togetherness in the changing room and I think that comes across in the games and to the fans and that's what we want. If this football club is ever going to achieve anything then that is what we want and it will be done by everyone - players and supporters - pushing in the right direction."
