Ajax scored twice within three minutes for a second successive win.
Content to contain the CAF Champions League title-holders for much of the clash, Ajax scored twice within three minutes for a second successive win.
When South African Roger de Sa quit as Ajax coach last month, the 'Urban Warriors' were bottom of the 16-club Premiership with only two points from a possible 21.
Menzo, who arrived in Cape Town this year to head an Ajax youth scheme, took over and has guided the team to two wins and a draw in four outings.
Few neutrals gave his youthful, star-less side a chance against a Sundowns team that travelled to Cape Town from Pretoria having won four consecutive league matches.
But after surviving several goalmouth scares and enjoying a few lucky moments, goals from Thabo Mosadi on 71 minutes and Ndiviwe Mdabuka on 73 minutes won the match.
Mosadi outpaced Thapelo Morena to fire a low cross past Uganda goalkeeper and 2016 Africa-based Footballer of the Year contender Dennis Onyango.
A dribble into the six-yard box set up Mdabuka to tap the ball over the line for the killer second goal at Cape Town Stadium.
"I am so proud of my team," said Menzo, 53, who helped former European champions Ajax Amsterdam win nine trophies and was capped six times by the Netherlands.
"The situation at the club was difficult and it is great to see hard work paying off. The emphasis has been on organisation and discipline."
Menzo was hailed as an "unbelievable coach" by Sundowns handler Pitso Mosimane, the first South African to guide a club to CAF Champions League glory.
"Let us give credit to Ajax -- they were the better team and deserved to win. We had chances to score before they did, but did not take them."
Sundowns host Free State Stars Saturday hoping to close an eight-point gap behind leaders Cape Town City, who have played four matches more.,
The Pretoria club then fly to Japan for a maiden appearance in the FIFA Club World Cup next Sunday.
"Keeping the minds of my players off Japan is a big challenge at the moment," admitted Mosimane.
Victory lifted Ajax out of the relegation zone to 14th, one point above Free State Stars and two ahead of bottom club Highlands Park.
Free State ended a 10-match winless run since the league kicked off last August by defeating off-form Platinum Stars 3-1 in central town Bethlehem.