Low cost cards are one of the interesting balance decisions in Hearthstone. The real cost of a card is its mana cost plus any built-in penalties plus the fact that it is a card.
As an example of the first two, Squirming Tentacle is a 2/4 minion with taunt that costs 3 mana, and Vulgar Homunculus is a 2/4 minion with taunt that costs 2 mana and does 2 damage to you when you summon it. There are many ways to have costs, and there is synergy in building decks that avoid costs.
The cost of the card itself is a reason why 1-mana cards are more powerful than you would expect. The mana is less of a cost than the fact that you are using up a card. It is an opportunity cost; those cheap cards are only great in the first few turns or to burn extra mana around your stronger cards. If you have played all your cards by turn 5, drawing a 1-mana card when you have 6 mana available is suboptimal. You get one card per turn, and you have seven spaces for minions on the board.
This means that fast, aggressive (“aggro”) decks will always remain fairly effective, like classic “weenie” decks in Magic the Gathering. There are fairly inexpensive counters to large numbers of cheap cards, but cards that are card-inefficient are likely mana-efficient, or else they are just trash cards you would never use. If I can get a 2/4 minion with a special ability for 3 mana and 1 card, I better do better than a vanilla 1/2 minion for 1 mana and 1 card. The only way to keep cheap cards from being overpowered for their mana value is to make them underpowered for their card value.
The full cost of a card includes the card itself. This adds value back to cards that include a draw, although deck-thinning in that way can be a cost or a benefit depending on the context. In Dominion, cards that provide a card plus an action plus something else are popular because they have no opportunity cost that turn; the card refunds both the card and the action you used to play it. But the “plus something else” is usually pretty small unless the card is expensive, and there is still the opportunity cost that you used up the buy for a turn on that Village, when you could have got a Smithy.