Returning after an absence of almost five years, 39-year-old AC Milan striker Ibrahimovic became the oldest player to represent Sweden, beating the record of 38 years and 59 days previously held by goalkeeper Thomas Ravelli.
Sweden went ahead 10 minutes before the break when Ibra chested down a ball into the box and volleyed it back at a tight angle for Claesson to take a touch and bundle home.
With the Swedes often careless in possession, Georgia fashioned a few decent chances and the best of them fell to substitute Levan Shengelia in the 82nd minute, but Kristoffer Nordfeldt made a superb save to keep his clean sheet intact.
Ibrahimovic failed to add to his tally of 62 international goals before being replaced by Robin Quaison in the 84th and adjourning to the bench to watch his side hold on for the win.
"It felt good. It felt like it was my first international match. It was a lot of adrenaline," Ibrahimovic told Sverige Radio.
"I think I could have done a lot more, but we won the match, and that is the most important thing," he added.