Long-standing distance marks fall in the King Baudouin Stadium as Mondo Duplantis and Faith Kipyegon fall short with world best attempts.
The Dutchwoman, who won 10,000m gold at last year's World Championships, surged clear of world marathon record holder Brigid Kosgei in the final minute to cover a distance of 18.930km.
Kosgei was later disqualified for stepping inside the track.
Mondo Duplantis cleared a meeting record of 6.00m in the pole vault but failed three times at a new outdoor world record of 6.15m.
Faith Kipyegon was also unsuccessful with her attempt at the 1000m world record inside the King Baudouin Stadium.
Sifan Hassan set the first world record of the night in the women's one hour.
An army of pacemakers did a superb job to take the field round well inside world record pace pretty much from the start, meaning the only question was who would finish in front.
Kosgei did her best to shake off the track world champion but at the gun signalling one minute to go, Hassan hit the front.
And despite Kosgei's best efforts to stay with her, Hassan - the world record holder over one mile - soon raced away.
Her final distance was 18.930km, over a lap further than the previous record of 18.517km set by Ethiopia's Dire Tune in 2008.
In her first official track race, Kosgei was disqualified for stepping inside the kerb with second place going to Kenyan-born Israeli Lonah Salpeter whose 18.571km was also further than the previous world best.
Hassan said in her on-track interview, "I'm very happy. I never thought I would run so far. Actually, at the start I didn't feel good but after 30 minutes it got easier. It was a very strong race. Thank you for organising this in a very difficult time. I'm so happy to be here."
Farah's record bid looked in serious doubt after 35 minutes when the last pacemaker departed leaving him alone with his training mate Abdi.
At this point, television viewers could see a hologram of Gebrselassie's world record run in lane three some 10 metres ahead of the pair.
But Farah, shouted on by Gebrselassie's former manager Jos Hermans, quickened the tempo to get in front of the Wavelight time guide on the kerb.
In his first track race for three years, Farah was doing most of the work at the front but Abdi took the lead with five minutes to go.
That move was clearly part of a plan as it saw the Belgian break Gebrselassie's 20km world record, unofficially timed at 56:20.02, and gave the four-time Olympic champion a welcome breather.
The Briton then produced a trademark kick to burst clear of his friend in the closing stages and record a distance of 21.330km, 55m further than the Ethopian great's mark from 2007.
Farah said on the track afterwards, "That's incredible. I'm very happy to break the world record today. Me and Bashir Abdi worked together. I'm so pleased for him and for myself. An amazing race to do it and to show the people what is possible.
"I feel tired but at the same time, in the middle part of the race we had to work hard and get on the light. We were just helping each other to get through it. It's nice to break a world record." - Mo Farah
Faith Kipyegon failed in her bid for the world 1000m record having come desperately close in Monaco three weeks ago.
The Olympic 1500m champion looked on course after 400m but running solo for the last lap was tough for the Kenyan who clocked 2:29.92, almost a second off double Olympic champion Svetlana Masterkova's 1996 mark set in the same stadium.
The Swede then had three unsuccessful attempts at a new outdoor world record of 6.15m, one centimetre higher than Sergey Bubka's mark of 6.14m from 1994, but only came anywhere near close with his first effort.
Jakob Ingebrigtsen won his last race as a teenager, running the last lap in glorious isolation as he took the 1500m in 3:30.69.
The Norwegian set a European record of 3:28.68 last month in Monaco behind world champion Timothy Cheruiyot.
He was unable to come close to that time tonight with no one able to stay with him after the pacemakers dropped out.
World heptathlon champion Katarina Johnson-Thompson had two season's bests in her two events of the night although she may think she could have done better.
With home favourite and reigning Olympic champion Nafi Thiam missing due to an Achilles injury, the Briton was slowly out of the blocks in the 100m hurdles on her way to fourth place behind Belgium's Anne Zagre.
Despite that, her time of 13.57s was her quickest of the year as Zagre won in 13.21s.
In the high jump, she equalled her season's best of 1.84m before three failures at 1.88m which left her down in sixth place.
Australia's Nicola McDermott emerged victorious with 1.91m.
Italy's Eseosa Desalu kept up his impressive recent form with a comfortable victory in the men's 200m in 20.39s.
Poland have strength in depth in the women's 400m and Iga Baumgart-Witan showed that here, clocking a season's best 52.13s.