Slovakia surprisingly top Group E after beating Poland while heavyweights Spain were held to a goalless draw by Sweden.
Victory over Sweden in their second game in St Petersburg on Friday would secure a place in the last 16 with a match to spare.
“It looks like maybe now we are the favourites because we are top but we are still underdogs in this group,” said former Napoli midfielder Hamsik, 33.
“That is what we have in mind and nobody will give us anything. Only with a good performance can we get something, like against Poland.
“We fought together as a team and gave it our best efforts. Many players ended up with cramps. It was all about the team spirit and we have to repeat that in another tough game ahead against a very difficult opponent.”
Hamsik has been playing his domestic football in Sweden since March after joining Gothenburg following a stint in the Chinese Super League.
He said: “From my personal experience in Sweden it is very typical for them to have fighting spirit, they are tireless. It will be very physical.”
Sweden midfielder Albin Ekdal sees Hamsik as one of the main dangermen.
“Hamsík was one of Napoli’s best players for many years and produced goals and assists,” said the Sampdoria player.
“He’s world-class. Though perhaps he’s not at his best now, he remains a player to be wary of and respect.”
Sweden coach Janne Andersson hopes his team will be able to express themselves more after a disciplined performance to nullify Spain.
Andersson said: “We put ourselves in a good position after the first round and now we need to build on that by playing a good match and hopefully getting a good result.
“The key will be to settle into our own way of playing. We had too little of that against Spain.
“I think Slovakia will not press as high as Spain and we will get to use other parts of our game.”
Slovakia defender Denis Vavro is unavailable after testing positive for Covid-19. The 25-year-old, who is asymptomatic, is now self-isolating. A member of the coaching staff also tested positive.