Damian Shirazi's second century of the competition helped Oxfordshire reach the quarter-finals with a seven-wicket victory over Shropshire at Challow & Childrey yesterday.
But they made hard work of chasing a meagre 212 to win their third Group 1 match of the season.
Oxon will now play in the last eight on June 14.
Banbury opener Shirazi made exactly 100 off 134 balls, including 13 fours, before being bowled by seamer Jono Whitney.
His innings was a model of control and came after he made 110 against Bedfordshire in his side’s opening match.
The hosts had looked home and dry at that point, despite never scoring at much more than four runs per over.
But the wicket produced some uncertainty and Oxon needed five off the final over.
A single from Shahbaz Ali off the first ball, followed by a confident boundary from Nathan Hawkes did the trick.
Earlier, Luke Ryan, a late replacement for Daniel Rowe, was Oxon's star with the ball, taking 4-26 in his ten overs.
Ryan's partnership with fellow spinner Paul McMahon proved decisive as Shropshire struggled to score.
Shropshire would have been hoping for at least 250 after winning the toss.
But they were immediately in trouble as Stuart Hole had Ed Foster caught behind for a duck in the first over.
Whitney, Shropshire's other opener, fared somewhat better, top-scoring with 54.
Rob Foster and Anthony Parker both made starts, but it was only a spirited 27 not out, including two sixes, from No 11 Ed Bayliss that took the visitors past 200 in the final over.
Hole and Richard Kaufman picked up two wickets each in what was an impressive team bowling display.
Oxfordshire began the chase in solid fashion, with Shirazi and Kaufman in control.
They reached 81-0 at drinks, but Kaufman fell to a spectacular diving catch from Whitney at gulley soon afterwards.
Oxon never looked flustered, although the run rate was not exactly pulsating.
Craig Haupt made a useful 31, as Oxfordshire seemed content to plod away rather than go for it, and their tactics could almost have backfired.
But they would certainly have taken victory with four balls to spare at the beginning of the day.
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