Erakovic went down 3-6, 6-4, 0-6 to the Croatian, ranked 189th in the world, who advanced to the grand slam through qualifying.
The New Zealand number one came back into the match after being broken three times to concede the opening set.
Her first serve percentage reached almost 70% in the second set as she forced a third and deciding set but Konjuh, last year's junior champion at the Australian Open and US Open, then took control of the match to reach the second round of a grand slam for the first time in her career.
It is the first time since 2010 that Erakovic, ranked 68th in the world, has failed to reach the second round in London.
Konjuh will face Belgium's Yanina Wickmayer in the next round after the former ASB Classic champion upset 17th seed Australian Sam Stosur 6-3, 6-4.
Erakovic will begin her doubles campaign later in the week.
