For more on the Chile mine breakthrough, go to CNN Mexico

Copiapo, Chile (CNN) -- The first of the 33 miners trapped underground for more than two months could be hoisted to safety as early as Wednesday, Chilean Mining Minister Laurence Golborne said Saturday.

He spoke hours after anxious family members and exhausted rescue workers reveled in joyful relief after a drill pierced the roof of an underground mine in Chile where the men have been trapped since August 5.

Sixty-five days after the mine's collapse, officials announced the drill had reached the miners as sirens blasted through the site to signal the achievement. Family members ran downhill, cheering and carrying Chilean flags.

What began as a "possible tragedy, with the help of God, is ending as a true blessing," said Chilean President Sebastian Pinera.

"The miners, their families, the rescue workers, the government and all the Chileans have shown unity, strength, of faith, of hope, that is recognized and admired by the whole world," the president said. "This shows that when Chileans unite for great causes, regardless how grand or ambitious they may seem, we are always able to reach our goals and conquer the highest peaks."

Still, despite the day's success, rescue workers continue to face tough decisions and hard work to get the men out of the mine.

"We have not yet rescued anyone and there is much ground to cover," cautioned the mining minister, who also told reporters that the families of the miners "are aware of the process still expected."

Later in the day, Golborne said he believes workers could begin to hoist the miners up by Wednesday. The rescue operation is likely to take around two days, he said, which would mean all 33 men could be pulled to safety by Friday.

Also Saturday, the miners successfully set off explosives in the mine to widen the rescue shaft through which they will be brought, Golborne said.

Mine engineers decided to encase just the top part of the shaft with steel tubing to prevent rockfalls and further collapses during the extraction process. Previously, they had thought they might need to encase the whole thing -- a process which would have delayed the rescue.

Raul Lyon, vice president of GeoTech, the company operating the Plan B rescue drill at the scene, told CNN, "I can't wait to see the first one to come out."

Lyon said the ordeal has been arduous and emotional for the workers, but "we had patience" and solved problems methodically as they came up.

"It was very exciting and also comforting that we were able to reach the tunnel after exactly 33 days of drilling," said Lyon, who noted that the miners are "happy" and "in good shape."

Rescue crews have been drilling three separate wider holes to send down the rescue capsule and hoist the men to the surface. Those rescue attempts have been labeled Plans A, B and C.

Brandon Fisher, owner of the Pennsylvania company that provided and oversaw the use of drilling bits for the Plan B team, felt a "huge sense of relief and joy" and noted that the success wasn't "in the bag until we were through."

"This project was unbelievably challenging," said Fisher, whose company, Center Rock Inc., aided in the rescue of nine miners who were trapped for more than four days after the 2002 collapse of the Quecreek Mine, also in Pennsylvania.

After the 2,300-foot hole in Chile is secure, materials and equipment will be brought in so that the rescue capsule -- dubbed the Phoenix -- can be lowered into the mine. Then, authorities will lower a doctor and a rescuer into the chamber, Health Minister Jaime Manalich said. Medical and rescue personnel will be in place to start extracting and treating the miners.

Once the men have been extracted, they will undergo about two hours of health checks at a field hospital set up at the mine. They will then be flown by helicopter to a hospital in the town of Copiapo -- approximately a 15-minute flight.

"You think we were happy, you should have seen those guys," said Gregory Hall, one of the rescue drill operators, after Saturday's breakthrough. "They were just going crazy. I mean, imagine, that's a big step towards coming home. So they were just going crazy. It was wonderful."

Miners who are healthy enough will be allowed to visit briefly with family members in a reunion area before being taken to the hospital, Manalich said.

Health officials have increased the miners' physical activity in recent days to gauge how they can handle the stress of being removed, the health minister added.

The overall response has been good, he said, though some miners have exhibited anxiety or have had minor cardiac issues.

The miners have been in contact with the outside world through a small bore hole that sends them food, water, supplies and other necessities.

Pinera said September 19, during his last visit to the site, that he planned to personally hug each of the miners as they were pulled from the rescue shaft. He is due to depart on a government tour to Europe in mid-October.

First lady Cecilia Morel Montes told reporters Friday night that her husband was in the south.

"He has a lot of obligations," she said. "Undoubtedly, he would like to be here permanently. But he also has to be permanently in a lot of other places and it's necessary to continue governing."

She said the crisis faced by the miners has brought Chileans together.

"They have become very familiar to every Chilean," she said. "All Chileans are equally expectant, equally nervous; there's a lot of empathy toward the families."

She added that the men's liberation will occur "as soon as possible," but only after every precaution is taken to ensure their safety.