The United manager exchanged gifts and admiration with Mr Piñera and Chile's first lady Cecilia Morel at a special ceremony in London and joined Viña Concha y Toro, the club's official wine partner, in inviting the miners to attend a match at Old Trafford.
Sir Alex said, "Manchester United prides itself on footballers with great courage. But the other quality we identify with in those 33 fantastic miners is endurance - endurance to survive."
The boss acknowledged and congratulated Mr Piñera on Chile's successful rescue operation, an event that captured the world's attention on 13 and 14 October when the miners were brought to the surface after 69 days of being trapped underground.
"It's due to everyone getting together, particularly the Chilean goverment doing their very best, that the boys were brought back up," said Sir Alex. "We're all delighted you're safe and we look forward to giving you a day to remember at Old Trafford."
In response, Mr Piñera drew parallels with United's history and the determination shown after the Munich tragedy in rising again to become Britain's first European champions.
"You were strong enough to stand up and continue fighting for those things in which you believe," he said. "It was not a case of being lucky, there was a lot of effort and team work that went into producing that great success in 1968."
Rafael Guilisasti, vice-president of Viña Concha y Toro, commented: "We pay homage to the