The African side won their opening two games at the event, against Oman and Australia, and needed just a point from the encounter against the Irish to make sure of progression.
Zimbabwe started their T20 World Cup campaign with a convincing eight-wicket victory over Oman after bowling their opposition out for just 106.
Sikandar Raza's charges then produced the upset of the global tournament so far with a 23-run win against Australia in Colombo.
The Aussies were on the brink of elimination already after losing to Zimbabwe and then falling to defeat against co-hosts Sri Lanka on Monday.
Mitch Marsh and his team still have one game remaining in the group against Oman, but that result will be academic as they cannot catch up to Zimbabwe, have five points compared to their two.
Zimbabwe will square off against Sri Lanka in Colombo on Thursday to see who will top the standings.
The African side are now guaranteed at least three further fixtures, against West Indies, co-hosts India and a continental derby against South Africa.
Yuvraj Samra hit a record-breaking century for Canada but New Zealand aced the run-chase to come away with an eight-wicket win to secure their place in the T20 World Cup Super 8s.
A sensational unbeaten century from Pathum Nissanka powered Sri Lanka to a superb eight-wicket victory over Australia in T20 World Cup action on Monday.
England were made to sweat for large parts but eventually showed their metal before beating Italy by 24 runs to book their place in the next round of the T20 World Cup.
Azmatullah Omarzai starred with bat and ball to help Afghanistan register their first win of their T20 World Cup campaign after beating UAE by five wickets with four balls to spare.
