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Kerber, who reached last year's final, has struggled for form in 2017 and lost in the first round of the French Open.
Though not at her best against world number 247 Falconi, the 29-year-old German came through 6-4 6-4.
Kerber will play Belgium's Kirsten Flipkens for a place in round three.
"The first round here is always tough," said Kerber.
"It was a good match from both of us and it is always good to have a difficult match in the first round. She had nothing to lose; she came from through the qualifiers."
Kerber won the Australian Open and US Open in 2016 and reached the final of Wimbledon, where she lost 7-5 6-3 to Serena Williams.
She initially showed glimpses of her best form on her return to Centre Court on Tuesday, racing into a 3-0 lead in the first set.
However, 15 unforced errors and an impressive display by Falconi meant Kerber had to work for her place in the second round.
Nine-time Wimbledon singles champion Martina Navratilova
Kerber said the memories of last year's final came back. She came out playing like she did last year in the finals.
She needs to get into that mindset more if she wants to get into the final again.
It was definitely a much better performance than we have seen so far this year. She still needs to be more aggressive but it was a step in the right direction.
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Kerber will lose her number one ranking if she fails to reach the final and one of those who can take it from her, Karolina Pliskova, breezed into the second round.
World number three Pliskova beat Caroline Wozniacki to win the Aegon International at Eastbourne last week and continued her good form with a 6-1 6-4 win against Evgeniya Rodina of Russia on Tuesday.
Meanwhile, it took Wozniacki three sets to see off the challenge of Hungarian Tímea Babos.
Former world number one Wozniacki, currently sixth in the rankings, won 6-4 4-6 6-1.
Pliskova will play Slovakian world number 87 Magdalena Rybarikova in round two while Wozniacki takes on Bulgaria's Tsvetana Pironkova for a place in round three.
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Russian Anastasia Pavlyuchenkova, the 16th seed, became the highest-ranked woman to be knocked out as she lost 3-6 7-6 (8-6) 9-7 against Australian qualifier Arina Rodionova.
Victory for Rodionova, who is ranked 166th, was her first singles win at a Grand Slam.
Agnieszka Radwanska, Wimbledon finalist in 2012, saw off the challenge of former world number one Jelena Jankovic to reach the second round.
The ninth seed from Poland edged the first set on the tie-break before breezing through the second to win 7-6 (7-3) 6-0.
France's Kristina Mladenovic, seeded 12th, defeated compatriot Pauline Parmentier 6-1 6-3 while Spain's 23-year-old Garbine Muguruza, seeded 14th, beat Ekaterina Alexandrova of Russia 6-2 6-4 to progress.
Former world number one Victoria Azarenka, who has recently become a mother, has complained about the "tough" schedule at Wimbledon after having to wait at the All England Club all day on Monday for her first-round match.
Azarenka, who reached the Wimbledon semi-finals in 2011 and 2012, played on Monday for only the third time since giving birth to her son Leo in December.
Her match was not scheduled for a specific court on the order of play, filed under 'to be announced'.
"I had to be here the whole day, which, for a new mum, is a little tough," she said.
"Hopefully I won't play like this again."
World number one Angelique Kerber avoided an early upset as she overcame spirited American qualifier Irina Falconi to reach the second round of Wimbledon.