Sodhi took four for 17, his best figures for Notts, as the Falcons fell away after looking on course to win their fifth straight match.
The visitors had been 93 for two, in pursuit of a victory target of 167, but Sodhi’s spell changed the complexion of the evening.
His second wicket had been that of Wahab Riaz, who blitzed his way to 53, his highest score in the format, making his runs from only 31 balls with four fours and four sixes.
Earlier, Tom Moores scored 48 as Notts completed their 100th victory in the domestic T20 competition, becoming only the second county to reach that landmark.
Moores scored his runs from 34 balls, with four fours and three sixes, as the defending champions made 166 for five after winning the toss and opting to bat first.
Dan Christian also weighed in with an unbeaten 39, sharing a stand of 73 with Billy Root
Lockie Ferguson was the pick of the Falcons attack, taking two for 15.
Derbyshire fell away after the fall of Riaz’s wicket and could only reach 157 for eight as the overs ran out.
Ferguson had put Notts under early pressure by bowling both Riki Wessels and Jake Libby but Moores hit Ravi Rampaul for consecutive sixes as the powerplay overs ended on 37 for three.
Samit Patel was dropped on 13 by Billy Godelman but couldn’t capitalise on his good fortune and only added nine more before Hardus Viljoen knocked over his off pole.
Moores fell to Matt Critchley’s first delivery but Christian and Root gave the back end of the innings some impetus.
Christian hit three maximums and Root also cleared the ropes as the Outlaws finished strongly.
Patel, who has featured in 96 of the Outlaws’ wins, removed Calum MacLeod in the opening over but Riaz and Billy Godelman made the chase seem a formality before Sodhi came on.
The New Zealand leg-spinner trapped Godleman lbw, had Riaz caught behind, bowled Gary Wilson and had Wayne Madsen caught in the deep.
Harry Gurney, with two for 33, ensured that Notts held on for the win, meaning that his side could celebrate an important two points and their place in history as the second team to reach the century milestone – getting there 13 days after Lancashire achieved the feat.