No team achieved nine points from a possible nine in the group stage as there were a number of upsets, most notably Argentina's loss to Saudi Arabia and Germany's first-round elimination.
Those shocks have continued into the last 16, with Morocco dumping out Spain on penalties on Wednesday (AEDT).
Speaking on the first rest day of the World Cup, Infantino said: "I have seen all the matches, indeed, and put very simply and very clearly, this has been the best group stage of a FIFA World Cup ever.
"So, it's very promising for the remainder of the FIFA World Cup.
"The matches have been of great, great quality in beautiful stadiums – we knew that already. However, as well, the public who was there was incredible. Over 51,000 on average.
"Record-breaking figures on TV – we already had over two billion viewers, which is really incredible.
"Two and a half million people in the streets of Doha and a few hundred thousand every day in the stadiums, all together, cheering together, supporting their teams, fantastic atmosphere, great goals, incredible excitement and surprises."
Discussing the surprise results, he said: "There are no more small teams and no more big teams. The level is very, very equal.
"For the first time as well, national teams from all continents going to the knock-out phase, for the first time in history. This shows that football is really becoming truly global."
Infantino expects the finals to conclude in a similarly entertaining manner.
"We hope that the FIFA World Cup continues and concludes as it has started – a fantastic success," he said. "I am sure that we will reach the five billion viewers around the world.
"When it comes to stadium attendances, the stadiums are sold out, full practically at every match. The fan festivals, the different fan zones, are also very crowded with people celebrating and enjoying [themselves].
"And at the end [of the day], we simply want to give some joy and some smiles to people all over the world. That's what football is about, that's what the FIFA World Cup is about, and that's what should also happen from now until the end."