Dunn impressed his manager Steve Bruce, who delighted to have started the season with three points.
Bruce said afterwards: "The second half was extremely difficult from our point of view. But we expected it to be from a side with so much talent as Spurs."
Tottenham, who finished last season in free-fall, took a long time to warm to their task and should have taken advantage of a wilting Birmingham side in the second half.
The Londoners staged a big improvement after Bobby Zamora replaced Helder Postiga early in the second half.
Robbie Keane had a marvellous opportunity to level the scores following a through ball from Goran Bunjevcevic, but he casually hit a half-hearted shot against the post.
The first half was a niggling affair which often continued into the second half. There were no less than nine bookings in the game with 45 free kicks.
The turning point occurred in the 36th minute when Savage won a penalty following a clash with Anthony Gardner. To many spectators it was a harsh decision as Savage appeared to have impeded the Spurs defender.
Dunn slotted home the spot-kick despite Kasey Keller getting a hand to the ball.
In the last minute of the first half Birmingham went close to adding to their lead when Geoff Horsfield sent Christophe Dugarry clear only for the Frenchman's cross shot to be saved by Keller.
Stephen Clemence had a very good game for Birmingham against his former club, while newcomer Helder Postiga could make little impression against Darren Purse.
The Portuguese striker failed to get in a single shot but in sharp contrast Zamora posed a lot of problems. The former Brighton striker put in several good goal attempts as Birmingham began to panic.