Mannarino started well but Del Potro's power game finally clicked and he gained the edge to move through 2-6 6-1 7-5 6-4.
Eighth seed Mardy Fish made it through in more conventional fashion with a 6-4 6-4 6-2 defeat of Luxembourg's Gilles Muller.
Fish spent many years in the shadow of countryman Andy Roddick but is now relishing his status as the American number one.
"The goal was always to get into the top 10," Fish said. "The goal was to do well in tournaments and ranking-wise, it never really occurred to me being number one in our country.
"But it feels good, it's a nice title and it's a nice position to be in."
Seventh seed Tomas Berdych beat Albert Ramos 7-5 4-6 6-2 6-3, Nicolas Almagro, seeded 10, edged out Lukasz Kubot 1-6 7-5 6-3 7-5, and 18th seed Feliciano Lopez saw off Leonardo Mayer 7-6 (7/5) 6-3 7-6 (7/2).
Bernard Tomic thrilled the Melbourne crowd by hitting back from two sets down to stun 22nd seed Fernando Verdasco on the opening day of the Australian Open.
The Australian number one, who won the warm-up event in Kooyong over the weekend, looked destined to make an early exit when he fell behind in a strangely lethargic opening before upping the aggression levels to advance 4-6 6-7 (7/3) 6-4 6-2 7-5.
Tomic is the highest-ranked teenager at number 38 and he displayed all his promise and plenty of nerve to outlast the experienced Spaniard on Rod Laver Arena.
"I don't know how I found the energy but the crowd kept me going," said the Germany-born Queenslander, who has the weight of a nation on his shoulders.
"It is hard to deal with (the expectation) but I learn to deal with it and I try to enjoy it.
"Today wasn't fun though, it was torture. I really don't know how I won but I am the happiest person alive right now."
Rafael Nadal stated his Australian Open intention with a straight sets win over American Alex Kuznetsov to breeze into the second round.
Nadal took a little bit of time to get into his work but duly advanced to the second round on Monday with a 6-4 6-1 6-1 victory over Kuznetsov.
The 10-times grand slam winner, who is battling a niggling shoulder injury and has said his knees are still a concern, took 44 minutes to take the first set against the persistent American.
But once he started to get into the groove, the 25-year-old left-hander ran away with the next two sets, dominating from the baseline and giving Kuznetsov just two break-point opportunities when he was serving for the match before clinching victory in 106 minutes with his ninth ace.
Nadal's long-time rival Roger Federer cruised through to the second round as well, with a straight-sets win over Russian Alexander Kudryavtsev.
The third seed admitted after the match he knew little of his opponent, having to check the internet at the weekend to ensure he was the player he thought he was.
But his lack of knowledge meant little on court, as Federer breezed to a 7-5 6-2 6-2 victory in little over an hour and half.
It was a welcome result for Federer who pulled out of the Qatar Open 10 days ago with a back injury.