Argentina beat Cape Verde 3-2 on Friday to book their spot against Egypt, who overcame Australia 4-2 in a penalty shootout earlier, in the next round.
But Argentine fans were off their seats as the Blue Sharks fought relentlessly right to the end before collapsing on the pitch after the final whistle.
"As this team has demonstrated many times, and as I've said many times, it competes," Messi said in Spanish. "And we competed to the end."
The winning goal for Argentina was credited as an own-goal after Cristian Romero's header deflected off Cape Verde's Diney Borges in the 111th minute.
Lionel Messi earlier scored in regulation to record his 20th World Cup goal and give La Albiceleste the lead.
Deroy Duarte then scored for Cape Verde before Lisandro Martinez netted in the 103rd minute to put Argentina 2-1 ahead.
But Sidny Lopes Cabral scored another great equaliser to stun the pro-Argentina crowd in South Florida.
Cape Verde's memorable World Cup debut had been a stunning run that few outside the tiny island nation off the western coast of Africa could have predicted.
Behind the stellar play of 40-year-old goalkeeper Vozinha, Cape Verde became the smallest country to reach the knockout round, securing surprise draws against Spain and Uruguay and another against Saudi Arabia.
"I have to give credit to our opponents," Argentina coach Lionel Scaloni said. "The truth is, when people say there's no such thing as an easy opponent, today (Cape Verde) proved they're a great team."
Vozinha made 10 saves against Argentina, including five against Messi.
"We dignify what our country is. We drew twice against the world champions, we took it to extra time," Cape Verde coach Bubista said.
"More than anything, it's being proud of our players who were dignified in the World Cup. We showed our identity."
Messi added another brilliant goal to his resume, sneaking behind the Cape Verde defence on a well-timed run as Martinez lofted a pass towards him, taking a quick touch and hammering a finish into the roof of the net in the 29th minute.
The Argentina captain has seven goals in this tournament - one more than France's Kylian Mbappé in the Golden Boot race as the top scorer in this year's World Cup - and extended his World Cup record, moving two goals ahead of Mbappé on the all-time list.
He has scored 12 goals during his record eight-match scoring streak.
MOST WORLD CUP GOALS:
20 - Lionel Messi (Argentina)
18 - Kylian Mbappe (France)
16 - Miroslav Klose (Germany)
15 - Ronaldo (Brazil)
14 - Gerd Müller (West Germany)
13 - Just Fontaine (France)
13 - Harry Kane (England)
12 - Pele (Brazil)
11 - Sandor Kocsis (Hungary)
11 - Juergen Klinsmann (West Germany)
11 - Cristiano Ronaldo (Portugal)
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