Vedad Ibisevic, who has been linked with a move to West Ham United, netted after just six minutes, and the goal looked to be enough to give them a half-time lead.
However, Christofi got his side back into the game in first-half stoppage time when he headed home at the near post.
After the break, Everton midfielder Muhamed Besic missed a great chance inside the box when he hammered a shot against a cross bar after evading two defenders with some neat footwork.
The Cypriots then had the lead when Christofi grabbed his second of the evening with just over 15 minutes remaining, although Miralem Pjanic spurned a great chance to earn a point in the final moments of the match.
The home side were awarded a late penalty, but the Roma midfielder missed his spot kick as the visitors held on to record a memorable win.