The pair played husband and wife Roy and Hayley Cropper in the ITV series for 16 years until Hayley's screen death from pancreatic cancer in January.
The storyline saw the well-loved couple work through Hayley's bucket list, which included dancing at the Tower Ballroom in Blackpool, before she took her own life with Roy beside her as her illness reached its final stages.
Neilson and Hesmondhalgh were awarded Best Actor, Best Actress, Best Dramatic Performance (Neilson), Best On-Screen Partnership, Best Storyline for Hayley's cancer and Best Single Episode for Hayley's death.
Hesmondhalgh said of not working with Neilson any more: "It's like a marriage. We went out for a pint in Manchester the other week, it was lovely.
"I told him all the gossip, even though I'm not in it any more I know all the gossip and he doesn't.
"That kind of working relationship comes once in a lifetime."
Neilson explained the toll the storyline had taken on him: "My personal life was a complete write off last year because I was living Roy's life."
He added modestly of the storyline: "I'm glad we got the result but I don't do pride much."
Corrie's other wins included Outstanding Achievement for Helen Worth who has played Gail McIntyre for 40 years, Best Comedy Performance for Simon Gregson who plays Steve McDonald and Sexiest Female went to Michelle Keegan for the sixth year running.
Keegan, whose character Tina McIntyre is due to be killed off this month, wore a black and white tux to the ceremony and said of her nomination: "It's flattering. I can't believe I won in the first place a few years back."
Hollyoaks picked up Best Soap and Danny Mac, who plays Dodger Savage in the show, held on to his crown as Sexiest Male.
The Channel 4 soap also got Best Villain in Anna Passey who plays Sienna Blake, and Ellis Hollins won Best Young Performance as Tom Cunningham.
BBC soap EastEnders got just two awards - Best Newcomer for Maddy Hill as Nancy Carter and Outstanding Achievement Off Screen for company manager Carolyn Weinstein.
ITV's Emmerdale managed to convert just one of their nominations into a win with Spectacular Scene for The Woolpack's flood and siege, while BBC daytime soap Doctors went home empty handed.
Bringing the glamour to the red carpet at London's Hackney Empire were Hollyoaks actress Stephanie Waring in a floor-length yellow lace dress, EastEnder Shona McGarty going for Hollywood glamour with wavy hair and a figure-hugging burgundy gown, and Corrie's Catherine Tyldesley who wore a hot pink full-length dress.
EastEnders' big signing, Danny Dyer, also made his Soap Awards debut looking dapper in a silver-grey three-piece suit, while Corrie's Bill Roache walked the red carpet ahead of his forthcoming return to screens as Ken Barlow.
The British Soap Awards airs on ITV today at 8.15pm.