The Reds went into the match level on points with the bottom two but climbed to 18th position after racing into a 2-0 lead at half time.
Bowerman, recalled to the line-up ahead of Danny Webber, put Stanley in front in the 15th minute when he headed in a cross from Nicky Hunt.
The striker then turned provided two minutes before the interval as skipper Joyce scored a rare goal - and an even rarer header.
Veteran Jamie Cureton pulled a goal back for Cheltenham in the 72nd minute but Stanley had done enough for victory.
Stanley: Marcus Bettinelli, Nicky Hunt, Dean Winnard, Tom Aldred, Laurence Wilson, Kal Naismith, Luke Joyce, Peter Murphy, Piero Mingoia (Will Hatfield 75), George Bowerman (Danny Webber 72), Kayode Odejayi (Rob Atkinson 87). Subs not used: Michael Liddle, Shay McCartan, James Gray, Josh Windass.