Michael Olunga opened the scoring, with Erick Omondi adding a spectcular second to send the watching faithful at the Kasarani Stadium in Nairobi into raptures.
The noise level was ramped up again on 67 minutes as Wanyama struck from the spot to round off the win and ensure Kenya's spot at the 2019 Africa Cup of Nations, which will take place in Cameroon during June and July next summer.
Kenya top Group F with seven points from their four games, ahead of Ethiopia, Ghana and Sierra Leone.
It wasn't all good news for Kenya and Wanyama, though, with the powerful midfielder limping off in the second half amid concern over a hamstring injury.