Warner failed to fire after scoring 70no and 71 in his previous two knocks, falling for just five off six balls after the Spirit had been inserted at the Aussie veteran's new Lord's home on Thursday.
But England star Jamie Smith (52) and New Zealand great, skipper Kane Williamson (45), dragged the home side back on track with a fine second-wicket partnership.
And another Australian Ashton Turner, who's made useful contributions throughout the tournament, chipped in with 30 off just 18 balls, featuring five boundaries, before Stoinis (2-27 off his 20 balls) got him caught at cover.
It helped the Spirit to a decent 5-162 off their 100 balls and though Rockets' opener Tom Banton looked in fine form again with his 46 off 31 balls, none of the other visiting batters ever looked comfortable at the iconic venue as even Joe Root struggled a bit for his 27 off 24.
Anglo-Aussie Dan Worrall was effective as usual, grabbing a couple of key top-order wickets, leaving Stoinis already under pressure when he got to the wicket at 3-86 with 60 balls already eaten up.
Starting too slowly to keep up with the required run rate, amassing just five off his first 10 balls, Stoinis did eventually explode into action, ending up with three sixes off his last seven balls in his 35no off 22 balls, but it was not enough to stop the Rockets limping to only 6-141 and falling to a 21-run defeat.
