Sweden's Anna Nordqvist held her nerve to end a four-year winless drought and claim a third major title after a dramatic finale to the AIG Women's Open.
Sagstrom bogeyed the 18th to miss out on a potential play-off but it was Denmark's Nanna Koerstz Madsen who had the most cause for regret, the 26-year-old sharing the lead with playing partner Nordqvist until making a double-bogey six on the last.
On the hole where Jean van de Velde squandered a three-shot lead in the 1999 Open, Madsen pushed her approach into a greenside bunker and, from an awkward lie, shanked her next shot almost out of bounds.
The resulting double bogey dropped her into a tie for fifth alongside Evian Championship winner Minjee Lee, who had earlier set the clubhouse target on 10 under after a closing 66.
Nordqvist, whose last victory was also a major, the 2017 Evian Championship, said: "I've been waiting on this one for a while. There's been a lot of downs, a lot of hard times, so I think this makes it feel even sweeter.
"I could only dream about winning the British Open and I'm so happy. I had my husband Kevin there for me every single year since I won last time. We officially got married in Scottsdale in March with just us and six of our friends there and about 21 people on Zoom.
"We have postponed what I would say is our real wedding to next summer. It's going to be about 20 minutes away from here in a castle. I can't wait to walk down the aisle again in Scotland next year."
Nordqvist and Madsen shared the lead heading into a final round which saw a six-way tie in the early stages before birdies on the sixth, eighth and ninth edged Nordqvist in front.
A two-shot swing on the 12th, where Nordqvist made her only bogey of the day and Madsen a birdie, gave the latter the lead for the first time, only for the 26-year-old to bogey the 15th after finding sand off the tee.
Sagstrom's birdie on the 17th briefly made it a three-way tie at the top, but she bogeyed the 18th after her drive trickled into a bunker, leaving the final pair to fight it out on the closing stretch.
Madsen suggested she had paid the price for a defensive approach to the 18th, saying: "I was nervous all day. I did really well in trying to still hit good shots and on 18 I tried to not make a mistake and that was the only thing I shouldn't do.
"I'm really proud of what I've been working on and I do believe that it's going to come in the future."
Hall's chances of a second title looked slim when she followed an eagle on the sixth and birdie on the seventh with consecutive bogeys, but she bounced back to birdie the 11th and chipped in for a second eagle of the day on the next.
A two-putt birdie on the par-five 14th took Hall to 11 under and four pars over the tough closing stretch saw her take over the clubhouse lead from Lee.
"Whenever you're contending in a major it's good," Hall said. "I'm very happy to have had a chance to win and it's very special for it to be at Carnoustie. I'm very happy with the way I played, so that's all I could ask."
Scottish amateur Louise Duncan was within a shot of the lead after a birdie on the first, but the 21-year-old Stirling University student eventually had to settle for a tie for 10th following a 72.
"It was outstanding," Duncan said of her overall performance. "Pretty surreal and just dead exciting. I feel a bit mind blown as if it's not actually happened yet. I'm glad it's not a fluke, that I played well all four days."