Messi has fired in decisive goals for Alejandro Sabella's men at the FIFA World Cup so far, netting the second goal in their 2-1 win over Bosnia-Herzegovina before finding a way through a steadfast Iran defence late on in a 1-0 result.
The Barcelona star's goals have assured Argentina of a spot in the knockout stages, although they must avoid defeat against Nigeria on Wednesday to finish top of Group F.
D'Alessandro, who played 28 internationals for Argentina between 2003 and 2011, said an over-reliance on three-time Ballon d'Or winner Messi could be the nation's downfall.
"The pressure he (Messi) already has, the common pressure, as he is the best of the world. He has already has got that natural pressure," D'Alessandro said.
"He knows he represents a lot to the group and to our country.
"He had a good game in the first match as he scored. In the second match he decided it with his goal.
"The important thing is not to be dependent only on him. Argentina was dependent on him in the first game. When a team depends on only one player everything gets more complicated in the last 16."
D'Alessandro said, while the degree of difficulty will increase for Argentina in the knockout phase, he believes they will improve.
"I believe Argentina will get better in the third match and it will continue growing and getting better in the last 16," the 33-year-old Internacional man said.
"The knockout stage will be more difficult. There they can play against more difficult teams, more important teams, so they need to improve."
D'Alessandro said improvement will come given the reaction to Argentina's narrow win over minnows Iran.
"Criticism from the media, and the people in general, showed that Argentina didn't play well," he said.
"I think this is good, mainly because the group need to understand that they need to improve a lot.
"It is needed to improve not only to qualify because they've already got it not playing so well as we fans expected.
"The qualification that was the main focus and it has been achieved.
"As the last 16 begins (so does) another World Cup. It is different, the knockout stages.
"I believe the criticism came because we expected a bit more and the Argentinean media expected a better Argentina in the second game not in the first.
"As I said before it is normal to be nervous and anxious. But in the second we expected the team to grow and that wasn't seen."
