Abbott claimed 4-14 from his 20 deliveries - the 10th best figures in the history of the Hundred men's competition - as the Phoenix restricted their hosts to 7-127.
The Australian allrounder was also there at the finish as his side snuck home with a ball to spare as Birmingham claimed their opening win.
Asked to bat first, the Spirit slumped to 5-30 after Tim Southee trapped Daniel Bell-Drummond lbw with his first delivery before Abbott saw off Michael Pepper, Adam Rossington, Dan Lawrence and Ryan Higgins in the space of eight balls.
Shimron Hetmyer briefly flickered but fell for 16 after top-edging Benny Howell to Adam Milne before Liam Dawson (36) and Andre Russell (37 not out) mounted a recovery.
Russell hit four sixes in a typically brutal knock which included three maximums off the final set of the innings, bowled by Milne.
The Phoenix also struggled against the new ball, slipping to 4-20 after Dan Worrall claimed two early breakthroughs including the valuable wicket of Moeen Ali for a duck.
Liam Livingstone was unlucky to be run out by Olly Stone at the non-striker's end before youngsters Dan Mousley (39 from 38) and Jacob Bethell (43 from 27) took charge.
The visitors stuttered when both players were dismissed with the finishing line in sight but Howell, who made an unbeaten 24 off 12 balls, held his nerve to smash the penultimate ball of the match over extra-cover to seal the win with man-of-the-match Abbott at the non-striker's end.
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