The Nashville Predators, the third team in the deal, received a 2018 fourth-round draft pick from the Blue Jackets.
Letestu had served as Edmonton's fourth-line center since 2015, when he signed a three-year deal with the team as a free agent. He has been reliable in faceoffs and on the Oilers' penalty-kill unit.
"Mark is a very good player who will bring versatility and experience to our lineup. He can play on the power play, kill penalties and is very good in the faceoff circle," general manager Jarmo Kekalainen said in a statement. "He is also a person of very high character who made an impact on and off the ice during his previous time here and we are very happy to bring him back to Columbus."
Letestu, a nine-year NHL veteran, played three-plus seasons in Columbus from 2011 to 2015.
He has tallied eight goals and 11 assists in 60 games this season. His career bests of 16 goals and 35 points came in 2016-17.
Letestu, 33, broke into the league as an undrafted free agent with the Pittsburgh Penguins in 2007. He made his NHL debut in 2009.
Aberg, 24, has played in 52 career NHL games, with three goals and seven assists. He has two goals and six assists in 2017-18.