"Evil always wins because it stops at nothing": A phrase or expression to reflect that ideology

A word that fits the description is "opportunism".

(OALD) opportunism The practice of using situations unfairly to gain advantage for yourself without thinking about how your actions will affect other people
♦ political opportunism

(SOED) 1 The adaptation of (political) policy, actions, or judgement to circumstances or opportunity, esp. regardless of principle; gen. the seizing of opportunities when they occur. Late 19th century.

(Merriam-Webster) Definition of opportunism:
the art, policy, or practice of taking advantage of opportunities or circumstances often with little regard for principles or consequences

It is the part "esp. regardless of principle" that allows to characterize this attitude in life as one of not being held back by the perspective of the evil there might be in a given action.

The following gives the impression that people indulge in opportunism because they have the feeling that it works.

(Wikipedia)

Some people regard an opportunist stance positively as a legitimate choice. Thus, the British Conservative statesman Stanley Baldwin is supposed to have quipped:

"I would rather be an opportunist and float than go to the bottom with my principles around my neck" – Stanley Baldwin

In opportunism, life is viewed as presenting "an endless series of opportunities", where the pattern of one's responses defines their identity. It can also be viewed as striving to realize or express certain principles. However, the moral dilemma implied by opportunism concerns the conflict of self-interest with the interests of others, or with following a principle: either to do what one wants or to do "what is the right thing to do". Thus, substantively, opportunism refers to the acting on opportunities in a self-interested, biased or one-sided manner that conflicts or contrasts in some way with one or more general rule, law, norm, or principle.

How about a converse sentiment described by the aphorism Nice guys finish last.

People who are decent, friendly, and agreeable tend to be unsuccessful because they are outmaneuvered or overwhelmed by others who are not so decent, friendly, or agreeable.

[Wikitionary]

You can actually use the word "Machiavellian", as Wordnik defines it:

Attempting to achieve their goals by cunning, scheming, and unscrupulous methods.

The fairly well-known aphorism might makes/is right fits the bill.

Those who are powerful can do what they wish unchallenged, even if their action is in fact unjustified.

He believed that might was right and woe betide anyone who stood in his way.

[Lexico]

If you're not cheating, you're not trying (various attributions, including here). If the goal is to win, not to be good, those who are willing to do unethical things will be more likely to win.

History is written by the victors (various attributions, interesting review here) Worry about winning first - when you win, you can always justify your actions after the fact. Thus, having to do unethical things should not deter you from winning.

There is no good and evil, there is only power and those too weak to use it - Lord Voldemort. Anything that deters you from winning, any hesitation, is a weakness - including thinking in terms of good and evil.

And, as answered by neo-logophile, Nice guys finish last - because winning requires ruthlessness.