The Irons, who last won against Chelsea on December 1, went ahead through a Mark Noble penalty after just three minutes when referee Mark Clattenburg spotted a shirt pull by defender Ryan Bennett on Winston Reid.
Norwich midfielder Robert Snodgrass rattled the woodwork with a free-kick before Joey O'Brien crashed home a second goal after 26 minutes, which proved enough for a morale-boosting victory ahead of next weekend's FA Cup tie against Manchester United.
Russell Martin netted a late consolation with a well-taken volley for his third goal in two games.
"I was wondering if we were going to hold our confidence and performance level at home," said Allardyce, who felt the referee had a "great view" of the penalty and made the "correct decision.".
"It has been really good even though we have had two back-to-back defeats but they showed they have the mentality to cope with whatever is thrown at them."
Allardyce hopes to have striker Andy Carroll available by the end of the month.
He continued: "The disappointing thing was the chances the front lads missed, with balls just flashing across the box needing a tap in.
"We lost a goal in the 90th minute which made us edgy in the end when we should have been out of sight.
"We will continue to try to brush up on our finishing techniques with the lads and hope the next time round we are more clinical with our finishing."