The Poland under-21 global, who has scored seven goals in 17 games this season, has signed a four and a half year contract with the Scottish Premiership champions.
Rocking into Paradise like...
That was made clear on Sunday evening when the striker landed in Glasgow to begin the process of closing the deal.
However, with the William Hill Scottish Cup tie away to Partick Thistle on Saturday on the horizon, the 21-year-old said: "I don't expect to be in the first team after three days".
"I understand how massive a club Celtic is, but I feel I'm ready for this now", Klimala told Celtic TV.
Delighted to have signed for Celtic, Klimala, who is aware of the massive size of the club, has expressed his desire to score a lot of goals for the side and give the fans joy.
"The past 24 hours have been intense". I waited for the final decision so that I could finally come here and meet everyone. I had the chance to see the stadium and I felt the atmosphere.
"I have come here to win a lot of trophies with the team and do my best, as I said before, to bring a lot of happiness".
Klimala started her career with her local club, Lechia Dzierzoniow, but an unfortunate fate at the Legia Warsaw Academy ended with her release. Celtic is a huge club with a lot of history.
"In terms of the pressure here, I know the fans at Celtic support the club and that is not going to overwhelm me", he said.