Federer went into the tournament as the favourite to pick up a first Singles gold medal after his Wimbledon triumph.
His defeat of Andy Murray three weeks ago took him back to number one in the world.
The 30-year-old, who won Doubles gold four years ago for Switzerland with Stanislas Wawrinka, fought back from two sets to love down against Falla in the first round at Wimbledon two years ago.
Federer looked completely in control at a set and a break up and had three match points at 6-3 5-3 to wrap up what would have been a textbook victory.
However, he could not take them and from there things got a lot more complicated. Falla broke back to make it 5-5, saved three more break points and then incredibly broke again to win the set.
It was some turnaround and the Colombian looked like he might be able to cause a huge upset when he retrieved another break at the start of the decider.
But Federer dug in, broke again to lead 4-3, and this time there was no way back for Falla, the Swiss player converting his fifth match point to triumph 6-3 5-7 6-3.
Next the world number one will meet another of the protagonists in one of his Wimbledon dramas, Julien Benneteau.
Federer trailed the Frenchman two sets to love in the third round of Wimbledon four weeks ago before turning things around.