Jack Draper became the first British man to reach the semi-finals of the tour with a win over Ryan Peniston.
The world number108 fought back from a break down in the second set to win.
The last British player left in the draw will face the Frenchman in the semifinals.
The top seed and British number one lost to Petra Kvitova.
Four Britons are going to compete at Wimbledon next week.
After the match was delayed due to bad light, Peniston looked fatigued, having completed a 6-7 (10-8) 3-6 6-3 win over Spain's PedroMartinez earlier in the day.
Despite being tired, Peniston broke Draper in the first game of the match and went on to win.
A break late in the set allowed Draper to serve the opener out, but he quickly found himself 7-1 down to the player who beat him at the Surbiton Trophy.
After staving off three break back points, Draper was able to take the fourth and force the match back on serve.
The team won 13 of the final 15 points to get to the semi-finals.
The first set was close and Norrie was disappointed to let it go.
The British took a break before the Frenchman hit back with his serve-and-volley technique.
Norrie served to take the set into a tie-break but a long pass brought up set point, before an unfortunate gaffe off the net cord gave the opener.
Norrie had the first break-point opportunity of the second set, but it was saved by a Cressy volley.
He saved three match points after going 0-40 down on serve, but he couldn't stop the fourth conversion.
Despite Kvitova's struggles this season, Dart was always going to have a hard time against her.
The big-hitting left-handed pitcher reached her first semi-finals of the season with an 8-7 victory over Dart.