Declan Hughes scored either side of a Ross McNeil goal for the Lowland League leaders in a 3-1 win over BSC Glasgow.
Ajax's 1971-72 run was at the highest level of competition in the Netherlands so remains the official world record.
Hereford's run came to an end in January of this year.
The Bulls - formerly Hereford United, who were wound up in December 2014 - were playing in the ninth tier of English football, while the Lowland League is the fifth level of senior football in Scotland.
Ajax, the Amsterdam club who have won the European Cup four times, sent two members of their club TV department to cover East Kilbride's match.
The visitors from the Dutch capital delivered cases of beer to the winning side and a video message of congratulations from Edwin van der Sar, the former Netherlands and Manchester United goalkeeper, who is now a marketing director at Ajax.
East Kilbride manager Martin Lauchlan: "It was relief because I don't think we played particularly well - I'll need to have a word with the boys about that.
"But we're absolutely ecstatic to get 27 wins; I would say it's the biggest day of my career. It's the biggest achievement I've had but I don't want to take credit away from the players."
On the presentation from Ajax and message from Edwin van der Sar:
"It was a special moment and a touch of class. Ajax have been very respectful of us. They have tweeted a few times, but to send over 27 crates of beer and two bottles of champagne just makes it special and I would like to say a great thanks to them.
"I'll be expecting a couple of flights over to Amsterdam to thank Edwin! But absolute quality for a world superstar to recognise East Kilbride.
"It's a special day. I don't really know what to say because I'm overwhelmed with that."
East Kilbride surpassed Ajax's run of 26 successive wins and matched the record of 27 set by Hereford - watched by representatives of the Amsterdam giants.