The Tigers marked their first home Premier League fixture back in the top flight with a win courtesy of Robbie Brady's first-half penalty on Saturday afternoon.
But a win looked unlikely just five minutes later when Ivory Coast striker Yannick Sagbo was given his marching orders for an alleged headbutt on Norwich captain Russell Martin.
Despite their numerical disadvantage for over an hour, Hull clung on and limited Norwich to long-range efforts before surviving six nervy minutes of injury-time to pick up three points.
Bruce said afterwards: "We've done fantastically well to hang on and defend for our lives really. The sheer will to hang on for the result was evident for everybody. The goalkeeper has made a couple of saves but they didn't really create the chance that they wanted, which we're delighted about.
"We need 10 wins to make sure we stay here (in the Premier League). We've seen glimpses today and it shows for me where the club has grown in the last 12 months. Our starting line-up today, no disrespect to the lads who started last year, is chalk and cheese, and if we can add and grow our squad, that's got to be our aim."
Speaking on the two contentious moments of the match, Bruce believes Michael Turner's challenge on Sagbo for the penalty was 'clumsy', while admitting he had no complaints with the red card that followed for his new signing.
Bruce said: "It's clumsy (Turner's challenge). We've seen them given and not seen them given. I've got no complaints (on Sagbo's red card), it was silly, a rush of blood, and he's done something stupid."
Former Tottenham duo Tom Huddlestone and Jake Livermore both made their full debuts for the Tigers and Bruce was full of superlatives for the pair after the match.
He added: "A lot is written about footballers. They could have stayed in the bright lights of London and picked up their money but they showed what good players they are today.
"They proved that they are really, really good players who can only get better. Around five lads made their home debuts today and they were all terrific."