Wales had waited 58 years to play at a major tournament, and it was a memorable return as goals from Gareth Bale and Hal Robson-Kanu gave them a 2-1 win in Bordeaux.
Bale struck his 20th international goal with a trademark free-kick after 10 minutes, but Wales had to dig deep after Ondrej Duda’s 61st-minute equaliser.
But substitute Robson-Kanu scuffed home the winner 10 minutes from time to hand Wales the perfect boost before Thursday’s Group B clash with England in Lens.
“We’ll enjoy these moments in this competition if we do what we did today,” said Wales manager Coleman.
“It’s not about coming here and enjoying it, and going home making excuses about it being our first tournament.
“There’s nothing to be afraid of, even if sometimes you are afraid of the unknown.
“We just have to go about our business and, if we do that, there’s every chance that we will get what we need.
“We know we’re up against it in the next game.
“We are the underdogs, England are a top 10 team, and they are used to high pressure situations. But that could work to our advantage, and if we are ourselves in can be enough to get what we want.”
Wales had arrived in France without a win in four friendly games since qualification was achieved last October.
But Coleman had promised before kick-off that Wales would rise to the challenge, and he felt his players did just that as they battled for the cause.
“Everyone will look at Gareth’s goal, but some of his best moments were at the end of the game at 2-1,” said Coleman.
“He used his intelligence. He was basically heading the ball out of play to waste time. It wasn’t pretty but he gave everything for the win for Wales.
“He knew it wasn’t about Gareth Bale, it is about Wales.
“You can see his passion and how much he feeds off our supporters – and that’s why three million people love him to pieces.
“We got the three points which is fantastic, but more important was the performance.
“When our supporters are like that – when you see that sea of red and hear them singing like that – it’s so important we give our very best, because after that there’s no complaints.”
Wales suffered a blow before kick-off when goalkeeper Wayne Hennessey had to drop out with a back spasm.
Hennessey’s late withdrawal allowed Liverpool’s Danny Ward to win only his third cap and make his first start.
“It was such a blow for Wayne and he’s a big reason why we’re sitting here,” said Coleman.
“He’s had it before and he has recovered quite quickly, so we’re hoping over the next 24 to 48 hours we will monitor it and see.
“Hopefully he can play some part on Thursday, but Danny stepped up and, to come in to that atmosphere and perform like that, I thought he was outstanding.”
It was Slovakia who almost scored from the game’s first attack when Marek Hamsik dispossessed Bale and left several Wales defenders trailing in his wake.
Hamsik had the goal at his mercy and drove his shot past Ward, but Ben Davies had managed to recover to produce a superb goal-line clearance.
It was a huge let-off for Wales but they were ahead not long after when a foul by Patrik Hrosovsky on Jonathan Williams nearly 30 yards out gave them a free-kick.
Bale lined up the kick and, although it was not in the corner of the goal, it still had too much power and dip to beat the outstretched hand of Kozacik. The goal threw Slovakia out of their stride after such a promising start and Aaron Ramsey forced a straightforward save from Kozacik. Wales were protecting Ward well and Slovakia wasted their own free-kick opportunity when Juraj Kucka fired over from 30 yards.
Bale almost scored a second when he met Allen’s cross and Kozacik had to scramble across his line to save by the post.
But Slovakia were taking a grip of proceedings and a double substitution paid off instantly.
Mak broke between the lines and he slipped a pass to Duda, whose firm left-footed shot gave Ward no chance to make it 1-1.
Ward was involved again when he had to beat away Kucka’s powerful effort with James Chester on hand to clear the danger.
Coleman responded with a double substitution of his own to stem the Slovakia flow, with Joe Ledley back five weeks after fracturing his leg and Robson-Kanu also sent on after injury.
Wales had altered the balance of play and Ledley’s pass allowed Ramsey to run at the heart of the Slovakia defence.
Ramsey seemed to stumble but the ball broke to Robson-Kanu and, although his shot was scuffed, it found its way through the legs of Jan Durica and past Kozacik.
Slovakia were not finished, though, and substitute Adam Nemec headed against the post four minutes from time.
Wales: Ward, Chester, A Williams, Davies, Gunter, Allen, Edwards (Ledley 68), Ramsey (Richards 88), Taylor, Jonathan Williams (Robson-Kanu 71), Bale. Unused substitutes: Hennessey, King, G Williams, Cotterill, Vokes, Collins,V aughan, Church, F Williams.
Slovakia: Kozacik, Pekarik, Skrtel, Durica, Svento, Kucka, Hrosovsky (Duda 60), Hamsik, Mak, Duris (Nemec 59), Weiss (Stoch 83). Unused substitutes: Mucha, Gyomber, Gregus, Sestak, Skriniar, Hubocan, Salata, Pecovsky, Novota.
Referee: S O Moen (Norway).