Uruguay for the first time since 1954, won two of the opening match at the world Cup, and Suarez became the first Uruguayan player to have scored at three different Federation Internationale de Football Association world cups.
"I do fancy them in this tournament", said Neville.
"The goal was special, because I wanted to make up for the other day against Egypt", Suarez said. "We have to prove we are at the same level of quality as they are".
"We have four points", Hierro said.
The World Cup continues when Uruguay takes on Saudi Arabia in their second game of Group A play on Wednesday.
Saudi Arabia had earlier lost their opener against hosts Russian Federation by 5-0 at Luzhniki Stadium a week ago.
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"We must recognise that he's a crucial player to our national team because he shows solidarity on the pitch, he has a sense of responsibility and he has that enormous capacity to score goals".
"We need to fine-tune some things, but we are more than proud because for three World Cups in a row we have advanced to the knockout round, so we want to enjoy this handsome moment", Uruguay coach Oscar Tabarez said. "Undoubtedly there were other factors aside from the pressure of playing in the opening game that made a difference, but it's true that the difference was enormous".
Facing elimination, the Saudis fought hard for a goal with Hatan Bahmi getting several early chances.
Strikers: Cristhian Stuani, Maximiliano Gomez, Edinson Cavani, Luis Suarez.
Tabarez's preference for old hands Sanchez and Rodriguez, who play their football in Mexico and Uruguay respectively, may raise questions about the next generation of players in the squad, which is comprised of 10 players aged 30 or more, double the number that reached the semi-finals in 2010. "We only have one thing in mind". "We were happier after the match against Portugal, but we only had one point".
Another drily noted: "Arab countries always differ in everything but agree to leave the World Cup together".
A lot of work remains to be done for Tabarez and his staff to ensure La Celeste return to their fluid best.