A used pitch gave both sets of bowlers plenty of assistance, especially the spinners, but Spriegel's innings of 47 was key after he had earlier opened the bowling and took two wickets for 21 runs. The hosts' 149 all out had looked substantially short of being competitive as Jade Dernbach (three for 39) and Gareth Batty (two for 26) also enjoyed success.
However, early wickets for Harry Gurney (two for 23), together with a tight spell from Steven Mullaney (two for 24) dragged Notts back into the contest, until Spriegel hit three fours and two big sixes in a 61-ball stay to help guide Surrey home with four wickets and 23 balls to spare.
It meant a happy return for Rory Hamilton-Brown, playing his first game since taking a break from action following the tragic death of team-mate and close friend Tom Maynard.
Visiting skipper Batty's decision to bowl first was soon proven a wise one as Notts were reduced to 52 for four by the 10th over. Michael Lumb was bowled trying to cut Spriegel, who also had Riki Wessels caught at short mid-wicket.
In between, Samit Patel top-edged Dernbach to long leg before the paceman returned to oust Alex Hales (25) as the opener drove loosely to Zander de Bruyn at point.
It was left to Adam Voges (36) and Chris Read (35) to rebuild the innings and they shared a stand of 67, but it took 18 overs to put together and produced just one boundary. Voges' demise - caught at slip by Gary Wilson off Murali Kartik's turning lifter - was the first of the final six wickets to fall for 30 runs in eight overs.
Surrey were immediately in trouble at three for two in their reply as Gurney claimed the wickets in successive balls of Hamilton-Brown and Jason Roy, the latter with a brute of a lifter.
Steven Davies (35 from 42 balls) and Wilson (29) calmed the visitors' nerves with a partnership of 60 from 14 overs, before the former skied Mullaney's fourth ball to Voges. Wilson followed, being stumped by Read off Graeme White, and De Bruyn also fell cheaply, but Spriegel kept Surrey ahead of the required run-rate.
The left-hander fell just short of fifty as he tried to smash White for a second successive maximum and was caught on the ropes by Wessels. But he had already done enough, leaving Zafar Ansari (14 not out) and Batty (10 not out) to complete the formalities.
In LV= County Championship action, Warwickshire ended day one of their clash with Middlesex top of Division One after picking up two bowling bonus points.
But Dawid Malan's 106 put the home side in a strong position on 324 for seven at Uxbridge. The Bears went into their game in hand level with Nottinghamshire on 135 points and Chris Wright collected three for 66 as they edged ahead.
Sussex skittled relegation-threatened Worcestershire for 162 inside 54 overs at Hove. Steve Magoffin took four for 34 as only Moeen Ali (43) and Richard Jones (30 not out) offered any resistance of note on an abridged first day. By the close, the hosts were 22 without loss from seven overs with Chris Nash 15no.
Rain also restricted the hours of play at Liverpool, where defending champions Lancashire's survival bid was also damaged as they slumped to 129 for five against third-placed Somerset. Ashwell Prince made 51 and put on 79 with Karl Brown (39) but Peter Trego took four for 34 to put the West Country side in control.
In Division Two, promotion-chasing Yorkshire were bowled out for a disappointing 249 against Northamptonshire at Wantage Road.
Lee Daggett took three for 64 and also ran out top scorer Phil Jaques for 75, while David Willey and former Tykes spinner James Middlebrook grabbed two wickets apiece.
There were contrasting fortunes for Essex and Gloucestershire after they won the toss and chose to bat against Leicestershire and Gloucestershire respectively.
Essex openers Tom Westley and Jaik Mickleburgh batted through the 57 overs possible at Chelmsford to finish with their partnership unbroken at 161. Mickleburgh struck 11 fours and a six on his way to 81 while Westley will resume on 75, having found the boundary eight times.
But Glamorgan crumbled to 64 for six as just 27 overs were possible in Swansea. David Payne took two for 10 in eight superb overs and Jack Taylor added two for 20, with Jim Allenby's 25 the top score for the rock-bottom hosts.