The Westmeath side now face Gallen CC in the semi-final after they overturned a 9-5 deficit at the break and turned around a game they looked like losing for extended periods.
Calasanctius took an early lead when excellent full-back Jonathan Kelleher landed the first of his three first-half penalties after 11 minutes.
Three minutes later Marist grabbed their first try after a clinical move involving Ben Carty and Fitzgibbon put Robert Henshaw in at the left corner.
But two further penalties from Kelleher, including a huge kick from halfway saw Calasanctius hold a deserved lead at the break.
Their impressive forward play — led by Sean O’Brien — continued in the second half and Calasanctius looked likely to hold on but Fitzgibbon struck with time running out and a splendid conversion from the sideline by Carty sealed the three-point win.
MARIST COLLEGE: R Henshaw; E Maher, C Seery, R Sheridan, C Fitzgibbon; B Carty, B Digby; B Teape, S Flanagan, S Meunargia; K Cooper, T Thornton; S Ryan, E Kelly, M Doddy.
Replacements: R Corcoran for Maher (46); C Langdon for Kelly (51).
CALASANCTIUS: J Kelleher; M Kavanagh, R Burke, P O’Donnell, B O’Connell; D Joyce, R O’Donnell; C Folan, S Dunne, E Breen; D Greene, B Huxford; P Hackett, E Grealish, S O’Brien.
Replacements: L Foland for B Huxford (47); C Considine for Breen (51); D Comer for Hackett (61); K Finneran for O’Connell (61); T Huxford for C Folan (61); M Daly for Kavanagh (66).
Referee: D Flood (CBIRFU)