The Slovaks rode a first period goal from Michal Kabac and a 34-save shutout by Denis Godla to a 3-0 victory over the Czech Republic in the quarter-finals of the world junior championship on Friday.
That gives the Slovaks a rare appearance in the semifinals, against heavily favoured Canada, and a shot at a medal.
"It's a big win for all of us," said Reway, the Slovak captain whose empty-net goal with nine seconds left to play helped end a seven-game world junior losing run against their chief rival. "For a small country like Slovakia to beat one of the best teams in the tournament, it was an excellent performance from the goalie and from everyone on our team.
"We don't go very far in tournaments like world juniors every year. Once every five years or so. People in Slovakia are proud of us and I'm sure they watched the game to the end."
Slovakia, which finished eighth at the last two world juniors, has only medalled once: a bronze in 1999 in Winnipeg.
Making the final four puts them in contention for more hardware.
"Nobody would expect us to be in the best four and now we're there and everything's possible," said Reway, a Montreal Canadiens draft pick who was the favourite of many of the 7,696 at the Bell Centre.
Reway, now with Sparta Prague in the Czech league, formerly played under Canada's head coach Benoit Groulx with the Gatineau Olympiques.
The Slovaks were looking forward to a rematch with Canada, who beat them 8-0 in the opening group stage game for both teams last week. Canada routed Denmark 8-0 later Friday.
"It can't be worse than the first time," said Reway before Canada's win over Denmark. "It was the first game for all of us and, for some guys, it's their first world juniors and they were nervous.
"I think from our first game to this one our performance is getting better and better, so we're different team right now. If we play against Canada, it's going to be a hard game for us, for sure, but if our performance stays as it is a this point I'm sure it can be evenly matched."
Added Godla: "We want to take back what they took from us."
Kabac got his first goal of the tournament 18:50 into the first period as his shot from close range went off goalie Miroslav Svoboda's stick into the air. It dropped behind him and slid over the goal-line with Czech forward David Pasternak swinging and missing the puck to try to clear it away. The goal was confirmed by video review.
The Czechs wasted three glittering chances in the second, starting with Ondrei Kase hitting a goalpost on a short-handed breakaway. Patrik Zdrahal fanned on a one-timer with the net wide open and David Kampf had a gaping net but saw the puck roll off the end of his stick.
Boston Bruins prospect Peter Cehlarik scored on a second crack at the puck after Czech defenceman Dominik Masin failed to clear it from the front of the net at 14:08 of the third. Reway added his fourth of the tournament into an empty net.
Slovakia outshot the Czechs 36-34.