Robert Snodgrass gave the Canaries a half-time lead, a Ben Foster howler gifted Grant Holt a second, the third was an own goal from former Ipswich defender Gareth McAuley and Jonny Howson scored a spectacular late fourth.
"I thought we were excellent from the first minute right through to the 93rd minute," said Hughton, following a third win in 20 games.
"Incredibly relieved. We've embarked on a tough season, we've had some good periods in that season. We've found it difficult since Christmas to get the right formulas to win us football matches. On a day that it really counted, it was arguably our best performance of the season."
Hughton had a fitful night's sleep ahead of the match and hopes for another on Sunday night in celebration of survival, which was his remit on his appointment last June.
"It means absolutely everything," said the former Newcastle and Birmingham boss.
"I'm not the most outwardly emotional person, but I'm chewing up inside. It's certainly my most satisfying (achievement), because of the pressure of the day. For this club, for this moment and the pressure on this game, this would probably be my proudest moment."
Hughton was able to savour the final 28 minutes after Foster's decisive error. The England goalkeeper met Wes Hoolahan's prod forward with a mis-kick which allowed Holt to walk the ball in.
"Once the second one went in, we started to believe that it could be our day," said Hughton, who felt his side were due some luck.
West Brom boss Steve Clarke was unable to explain his side's failings. He said: "It's a blow to us because that's not a normal performance for West Bromwich Albion. It was Norwich's day, not ours."
- Sports & Recreation
- Jonny Howson
- Gareth McAuley
- Robert Snodgrass
- Grant Holt