Marcos Rojo was the unlikely hero as Jorge Sampaoli's embattled side sealed second place in Group D at Nigeria's expense four minutes from time after Victor Moses had equalised from the penalty spot.
Much of the focus was on Messi, who brilliantly opened the scoring as Argentina dominated early on before the flaws apparent throughout a stumbling campaign again came to the fore.
Mikel and Messi first locked horns in 2005, when a pair of penalties from the Barcelona star gave Argentina a 2-1 win over Nigeria in the Under-20 World Cup final.
Senior World Cup wins at the group stage in 2010 and 2014 followed – with Mikel playing in the latter game where Messi hit a brace.
"I'm sad because I think it was the moment I came so close to getting one," Mikel told reporters at Krestovsky Stadium.
"It's not a competition between me and him. I just felt that this time we almost came through and it would have been amazing for us to go through.
"Once again, luck was on his side and he managed to come through again."
Despite their struggles in the tournament so far, Mikel believes Argentina can launch a convincing bid for the title that has eluded Messi. France are up next in a mouth-watering last-16 tie.
"They have a great team," he said. "I think they are going to do well. I think now they've gone to the next round they will be more relaxed, they will play better football.
"I think they are a great side but we managed to keep them quiet, we managed to do our job and I am so happy and so proud of the boys. We have to keep going."
Goalscorer Moses is not as convinced as his captain over Argentina's merits.
"They're a good team but it's not like they beat us 4-0 or 5-0," said the Chelsea wing-back.
"They'll find it hard. Whichever teams they play next are going to be strong so we'll see what happens."