The Edinburgh Western MSP, who was the only candidate for the top job, was first elected to Holyrood in 2016 and was most recently the Lib Dem spokesman for health and sport at Holyrood.
His ascension to party leader comes after Mr Rennie last month announced he was standing down after ten years.
Lib Dem national leader Sir Ed Davey, tweeted: "A huge congratulations to @agcolehamilton on his election as Leader of @scotlibdems.
"Alex is a true liberal and tireless campaigner. I look forward to working closely with him to fight the climate crisis, value carers, and deliver world-class education and services for Scotland."
The Scottish Lib Dems lost one seat in May's Holyrood election, leaving the party with four MSPs.
However, Mr Cole-Hamilton secured the highest number of votes of any candidate in the history of the Scottish Parliament, with 25,578 people casting their constituency ballot for him.
Christine Jardine, Lib Dem MP for Edinburgh West, announced Mr Cole-Hamilton's success at a party meeting on Friday.
She told the crowd: "He's not just someone who's in this because of politics. He's in this because of what he believes and it will be his vision, his views, his direction that will take the party forward into the next phase.
"We all have a job to do in supporting Alex, we all have a job to do in getting the message out there, and we all have a job to do in ensuring that this next phase of our history is a strong, positive one.”
She added: "Alex is the one who's going to lead us forward in portraying that to the Scottish people, winning their support again, and making sure that we change Scottish politics."
Mr Cole-Hamilton, who was first elected to Holyrood in 2016, pledged to offer an “alternative kind of thinking” and to appeal to Scots “tired of having to choose between extremes” of nationalism.
He said he was inspired to enter politics while working in the children’s voluntary sector and the “fire lit in me [to improve lives] rages to this day”.