The original four-month ban for biting Italy's Giorgio Chiellini at the World Cup in June prevented him from training with Barca, but that part of the punishment was rescinded on Thursday by CAS, whose ruling also allowed him to take part in friendly matches.
Barcelona failed, however, to reduce the suspension.
"I am glad to feel like a player and be here with my team mates," the controversial Uruguayan striker, who joined Barcelona from Liverpool in July for a reported 75 million pounds ($125.19 million), told Barca TV.
"I felt I was in an awkward situation," he added. "I was paying for an error I made and had apologised for.
"Now it's time to forget about it, to leave all this behind, and to think about the future and the present and the fact that I am at this great club I always dreamed to play for."
Suarez will be introduced to the Barcelona crowd before Monday's home game against Mexican side Leon.
Neymar again trained with the squad on Friday following his return since fracturing a vertebra playing for Brazil in their World Cup quarter-final against Colombia on July 4.
The striker returned to light training on Monday although he has yet to be given a final 'all-clear' by the club's medical staff.