Zambia almost shocked the hosts moments later when Kelvin Mubanga pounced on a defensive error, only for Michael Kibwage to clear off the line.
Kenya responded strongly, with Ogam and Odhiambo both going close, while captain Aboud Omar's thunderous long-range strike flew narrowly wide.
Still, the visitors held firm, thanks largely to their goalkeeper Charles Kalumba, who pulled off a string of fine saves to frustrate the home side. The two teams went into the break locked at 0-0 despite Kenya's dominance in possession and territory.
The second half followed a similar pattern-end-to-end action with Zambia threatening through Prince Mumba and Mubanga, before Kenya regained control. Harambee Stars coach Benni McCarthy made key substitutions, bringing on Masoud Juma and Boniface Muchiri to inject fresh energy into the attack.
The move paid off in the 75th minute when Muchiri found space on the right flank and delivered a clever ball into the box. Ogam controlled with composure before slotting a low finish past Kalumba to finally break the deadlock.
Zambia fought hard for a response, with Mubanga testing Farouk Shikalo in the Kenyan goal late on. The Harambee Stars keeper was equal to the challenge, producing a superb save to protect the narrow lead.
At the other end, Muchiri and Marvin Nabwire both forced Kalumba into late saves, but Ogam's strike proved enough to carry Kenya over the line.
The win capped a remarkable group campaign for Kenya, who beat Morocco and DR Congo earlier and conceded just one goal in four matches.
For Coach McCarthy, the performance was about more than just advancing to the quarter-finals.
"When everybody said Group of Death, people thought we would have zero points after four games. But we are on top of the mountain now, and we really want to enjoy this view," he said.
Kenya's players credited the electric support at Kasarani for pushing them through a difficult match. Defender Daniel Sakari highlighted the role of fans in keeping the team's spirits high, especially in the tense closing minutes.
With Group A complete, Kenya finished top on 10 points, Morocco followed with nine, DR Congo ended on six, Angola managed four, and Zambia went home empty-handed.