Sibley, who has scored an undefeated 215 in the first innings, followed up with 109 as the visitors reached their target of 271 from a minimum of 58 overs with eight wickets to spare.
The 24-year old opener scored his runs from 147 balls, with 10 fours, and shared in an opening stand of 146 with Will Rhodes, who made 65.
Having been on the field for virtually the entire match, Sibley was eventually bowled by Paul Coughlin, who took two for 85.
By that stage the victory was already in sight, with Michael Burgess hitting an unbeaten 61 after coming in at number three in the order.
Earlier, the day had looked as if it might belong to Joe Clarke, who also registered his second hundred of the match.
Clarke followed up his first innings score of 125 with 112 as Notts made 260 in their second innings. He was last out, bowled by Oliver Hannon-Dalby, who took four for 54, with Jeetan Patel picking up three for 52.
Clarke’s championship season had begun with scores of 112 and 97 not out against Yorkshire in early April but off-field issues disrupted his campaign and he had failed to reach 50 in any of his 17 Division One innings since then.
Dropped for the previous match, he returned to the starting eleven as Notts rested players ahead of this weekend’s Vitality Blast Finals Day. His return to form now puts him in contention to figure on Saturday as well.
The former Worcestershire batsman reached his sixteenth first-class century from 130 deliveries, having hit 15 fours.
Resuming from their overnight score of 105 for five, a lead of 115, the home side had an early set-back when Coughlin was given out caught behind off Henry Brookes for 16.
Ravi Ashwin joined Clarke and kept the scoreboard moving in a partnership of 73, made in only 13 overs.
George Garrett broke the stand, gaining a positive lbw verdict against Ashwin, who made 42. The same bowler wasn’t able to build on his success, clutching his thigh as he limped out of the action during his next over.
Clarke was on 89 when he lost his next partner, Luke Fletcher, popping Patel into the hands of Sam Hain at short leg.
Patel dismissed Chappell for a promising 29 soon after lunch and the innings was completed when Clarke was bowled trying to farm the strike. In his disappointment he knocked out a stump with his bat – apologising immediately to the umpires.
Brendan Taylor, who made 114 and 105 not out against Durham in 2016 was the last Nottinghamshire batsman to record two hundreds in the same match.
The visitors began well, reaching tea on 82 without loss, with Sibley unbeaten on 48 and Rhodes on 34.
Gradual accumulation saw the openers advance the score to 146 before they were separated. Rhodes, who had hit Ashwin for two leg-side sixes in his 80-ball half-century, picked out Chappell in the deep, upper-cutting Coughlin.
A tired-looking Sibley then wandered across his stumps but Burgess reached his 50 from 51 balls, leaving Matt Lamb to hit the winning run.
The match aggregate of 1,517 runs is the largest in the competition this summer but leaves Notts still without a victory at the foot of the table. They now turn their attention to Finals Day on Saturday, before completing their red-ball season against Surrey at The Kia Oval next week.
Warwickshire move up to sixth in the table with their victory and host Yorkshire in their final encounter.