Sock was on the way to level the match after losing the first set 7-5, leading 5-1 in the second Mohamed Lahyani called a halt to proceedings.
The American had noticed some blood on one of the balls which transpired to have come from one of the ball kids, who had suffered a cut.
The match referee was called to deal with the problem which saw three new balls introduced into the action.
The switch initially favoured server Cilic, who emphatically closed out the game, but Sock won the set one game later to force a decider.
The American only qualified for the ATP World Tour Finals nine days ago when he beat Filip Krajinovic from a set down to win the Paris Masters.
But despite the fact he never played at ATP World Tour Finals, Sock looked to enjoy the unique atmosphere created at the O2 Arena.
“I thought the intros were pretty sweet,” Sock said.
“The floor was all lit up, pictures.
“I thought the graphics were really cool.
“I think it makes for a fun atmosphere. The fans love it.
“It’s fun to walk out to matches like that.”