Rangers went a goal ahead when Kevin Thomson surged into the Kaunas box to get on the end of a clever pass from Nacho Novo.
But a wonderful free-kick by Nerijus Radzius before half-time and a late Linas Pilibaitis header from a corner put Kaunas through to the final Champions League qualifier against Aalborg of Denmark.
Gers boss Walter Smith had no complaints afterwards: "It's always difficult to explain what went wrong. We just didn't play well. It was an extremely poor performance.
"We haven't started the season well at all. It's disappointing that we didn't reach any sort of level.
"It was one of those games where you never felt comfortable at all.
"It seems ridiculous that you can reach a European final and then get knocked out in the first game of the next season, but that's what's happened.
"Everyone who was involved is extremely disappointed. We've got no excuses.
"Kaunas were still well in the game at Ibrox. Tonight we had opportunities, not very many, but we had a few. We gave away needless free-kicks and corners."
Smith continued: "If it had been a situation where we were a new team that had been put together, then fine, you'd say that's something that could be expected to happen, but we're a reasonably experienced team.
"We never really handled either game particularly well."