Jordan, who took two for 32 and also ran out Spirit captain Eoin Morgan with a direct hit from mid off, defended 11 from the final set of five as the home side fell just short on 141 for seven.
London had looked set to break their duck in the tournament after an explosive opening stand of 68 from 27 balls between Adam Rossington and Josh Inglis.
But Jordan picked up the key wicket of Inglis, one of three catches for Quinton de Kock, as Spirit failed to chase down the visitors' total of 145 for six, built around Alex Davies' knock of 50.
In a result that blows the table wide open, the Phoenix recorded their second win of the competition while applying a brake to the Rockets' charge.
Trent Rockets claimed their second win after beating Birmingham Phoenix by 11 runs.
While the Rockets move on to four points, the result leaves the home side with much to do to achieve a top-three finish after suffering their third defeat in four games.
Bowling first, Birmingham ran into a late salvo from Heather Graham, whose unbeaten 44 off just 21 balls with two sixes and four more boundaries, lifted the Rockets to 145 for five.
The Phoenix's reply fell short at 134 for nine after a top-order collapse to 58 for five left the lower batters with far too much to do.