Federer, bidding for a record-equalling seventh men's singles title, was staring down the barrel as the south American world number 60 served for the match at 5-4 in the fourth set after producing a spellbinding display of baseline power and volleying guile.
"I definitely got very lucky out there," Federer said in an interview as he came off court. "I have lost many matches this year which I should have won, this is one I should have lost but I came through. But that is sometimes how grass court tennis works," he added.
The Swiss top seed, who also recovered from 4-4 0-40 on his serve in the third set, finally found his game after being made to look second best for much of the match before a disbelieving Centre Court crowd.
He wrapped up the fourth set tiebreak 7-1 and galloped through the decider in double quick time, avoiding the ignominy of being the first Wimbledon champion to fall at the first hurdle since Lleyton Hewitt in 2003.
Federer, who pummelled Falla for the loss of three games in the tune-up event in Halle last week, looked lost in the middle part of the match as the Colombian returned superbly, picked his advances to the net exquisitely and matched the 16-time grand slam winner stroke for stroke from the back of the court.
When Federer won the third, Falla could have been expected to fold but he broke in the opening game of the fourth, the Swiss contributing to his total of seven double faults in the process.
Falla could not hold his nerve serving at 5-4 and there was little doubt about the outcome once Federer took it into a fifth set, which he won in 27 minutes.
He will face Serbian qualifier Ilija Bozoljac in round two.