Adam Peters went over in the corner for the visitors, but Uili Kolo'ofa'i and Max Argyle scored to put Reds 14-7 up.
Richmond moved to within two points as Cameron Mitchell went in, before Lewis Robling scrambled over for the hosts.
Ali Chisholm scored to set up a tense finale but Ross Adair sprinted clear to end Jersey's season with a victory.
Harvey Biljon's side finished the campaign level on points with fourth-placed Doncaster, but missed out on the play-offs because they won fewer games.
Reds went into the match knowing they could not finish in the top four and changed their entire starting XV, after defeat by Doncaster last time out ended their play-off chances.
The Channel Islanders could yet finish the season with silverware, however, as they play Munster A in the British & Irish Cup final on Friday.
Jersey: Katz; Adair, Robling, Eastwell, Lane; Cope, Dudley; McCarthy (capt), Selway, Armstrong, Kolo'ofa'i, Campbell, Argyle, Voss, Rae.
Replacements: Macfarlane, Woolmore, Tampin, McKern, Haining, Fisilau, Ma'afu.
Richmond: Kirby; Simpson-Hefft, Mitchell, McLean, Woodward; Gabbitass, Stuart; Walford, Grimstone, Harden, Warden, Corker, Peters, Allcock, Davies.
Replacements: Townsend, Tarrant, Edwards, Gibbs, Chisholm, Sargeant, Cunnew.
Attendance: 2,210
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Two tries in each half helped Jersey Reds to a bonus-point win over Richmond which secured their highest-ever finish and points tally in the Championship.