Guzan stopped Robert Snodgrass' effort from 12 yards, harshly awarded for handball against Ciaran Clark, with Kozak scoring on the half-hour just seconds after replacing the injured Christian Benteke.
The missed penalty, just five minutes into the match, proved to the hosts' best chance and followed on from Villa's Andreas Weimann hitting the post in a lively start to the match.
Villa's American keeper proved to be the hero again in the closing stages, brilliantly tipping over a close range effort from Gary Hooper on 88 minutes to preserve his side's narrow lead.
The win ended a run of three consecutive defeats for Villa, but for Norwich there was a worrying lack of threat in their second straight loss.
Villa manager Paul Lambert returned to the club he left in 2012 amid lingering acrimony surrounding his exit.
Having been embroiled in a bitter fall-out with the Norwich hierarchy over his departure, Lambert had insisted in the lead-up to this fixture that he was not close to burying the hatchet with the Canaries board. Two wins at Carrow Road since his departure for the Villa boss added to an intriguing sub-plot.
After pressing the hosts from the whistle, Villa were cursing their luck after just three minutes when Weimann's expertly curled effort beat Norwich keeper John Ruddy, only to rebound away off the left hand post.
Norwich then had an even better chance to take the lead when Clark was harshly adjudged to have handled the ball inside the area. While the ball certainly hit the Villa defender on the arm, whether there was anything Clark could have done about it, having taken a deflection off Rocky van Wolfswinkel's head, was debatable.
However, Guzan came to the rescue, dropping smartly to his left to parry away Snodgrass' attempt from the spot.
Van Wolfswinkel elected to chest in a Nathan Redmond cross goalwards from close range on 12 minutes, but the Dutchman was left to rue his decision not to use his head when the ball flew over Guzan's bar.
Villa's Benteke limped off with a groin problem after the midway point of the first half, but replacement Kozak made an immediate impact.
The Czech striker had been on the field for just over a minute when he beat Ruddy at the second time of asking, placing a rebound from the home keeper's parry into the net for his first goal since moving for £7m from Lazio.
Shorn of Benteke's pace, Villa adapted their style to play with patience cushioned by their lead before and after half-time.
Guzan reacted quickly to push away a deflection off Leandro Bacuna, but despite his side asserting themselves more after the break Norwich boss Chris Hughton had seen enough from his starting 11, throwing on Gary Hooper and Anthony Pilkington.
Despite the positive replacements, Norwich became increasingly desperate, yet they failed to worry Guzan. Villa, content soak up pressure, hit on the break with Aleksander Tonev testing Ruddy from range.
Clark touched a Russell Martin cross dangerously close to his own goal, but the Irish international did much better seconds later to get a vital block on a Van Wolfswinkel cross.
With time running out for Norwich, left back Javi Garrido found Hooper with fine cross, but the former Celtic striker was left waiting for his first Norwich goal when Guzan flung a hand out to deflect the ball over his bar.
There was no way back for the Canaries after Guzan's outstanding save. Lambert left his former stamping ground not only with his third win in a row over his former employers, but with his side having kept a clean sheet for the first time in 27 Premier League outings, thanks in no small part to a fine display from their keeper.