Lenagan, previously a non-executive director, sold his software business WorkplaceSystems at the end of last year and will assume responsibility for the day-to-day running of the club with immediate effect.
He said in a club statement: "This move is a natural progression which had been planned to take place at the end of the current season, but which has been brought forward to coincide with Kelvin's recent decision to leave earlier to consider other opportunities abroad.
"I would like to put on record my thanks and appreciation for the excellent job that Kelvin has done as chairman and chief executive for the past four years.
"When I appointed him, the club was in a poor state across many areas and it is largely down to his efforts that Oxford United is in a better position now."
Lenagan, who has signed off on an increased playing budget for next season, added: "The aim is to leave League Two behind us. Chris Wilder and I are both clear about that and the players are clear about that. It won't be easy but if we all pull together, get our heads down and work hard, we can achieve the ambition that the players, the staff and the supporters all share. Promotion."