Yhyjyk

The protagonist sets out on a journey to reach a goal. But the further he gets into the story, the clearer he understands that the odds aren't in his favor. He experiences loss, frustration, anger, fatigue. He falls, rises again, and falls once more.
When he seems to be running out of option, thought, a stroke of luck™ appears.

Will this be accepted by the audience?

The idea here is that sometimes, casuality is in our favor. Sometimes good things do happen, so, by extension, they should happen in our stories, too.

Of course I'm interested in avoiding the Deus Ex Machina, were the resolution to a problem is delivered by the sheer force of will of the author alone. DeMs come with a set of drawbacks - first and foremost being very unsatisfying for the audience.

One of the best advice in avoiding DeMs is giving a proper foreshadowing to the readers - eg. introduce the elements that will eventually solve the dire situation before that the situation gets solved.

While I totally agree, I don't feel this is always applicable. Sometimes we are talking about random chance - no way to foreshadow that - and some other times the solution is being prepared outside of the narrator PoV.

In my novel, the MC has gone through a series of losses and she's rapidly losing hope. Somehow, I think that this will make any change of fortune more acceptable by the readers. The lucky event will bring to a resolution, but it won't solve all her problems; it will just point her in the right direction.

To sum up:

Is it possible to use luck (real or perceived) without making a Deus ex Machina? Does it help having the lucky event happen after a series of unfavorable ones?

I'll leave here a very related question that focuses on DeMs in general.

Related:

• How do you make random chance/happenstance not seem like deus ex machina?

Short answer: yes, with measure and forethought.

Note #1: unless karma and universal balance is a defining characteristic of your world, previous bad luck does not count. That's just how our world works.

Note #2: there are great examples of book that handle great moments of luck, and base their entire plot on it. I am guessing this is not the case in this question, and that the stroke of luck mentioned above is merely being considered as a lesser plot device.

First and foremost: keep luck reasonable.

Luck should serve merely as an input, or a very small step towards the MC goal. Be careful to not make it into the meaningful event that allows reaching the goal: while being great, luck has the unfortunate habit of removing the conflict with the world that you have built in your story. Too much of it and you throw a well crafted plot into a flat and dull series of pointless events.

For instance, if the MC is out of money, you could make them win the lottery with a ticket they find in the street, or you could get them to find a lost wallet with very little money in it. It is also true that luck can get greater as time passes, see the next section for this.

Second: make your MC deserve their luck.

This is also about conflict. If the MC does not try hard to get somewhere, then they probably do not deserve to get there. If it has to happen anyway, then see the next section.
From a reader's perspective: what am I reading if they struggle to achieve X but they get Y instead, which is great, but not what I have been rooting for?
You need to build expectation. The greater the expectation, the greater the relief when a bit of luck helps the MC. If your novel is a 800 pages book about a poor person trying to get through in any possible manner, then yes, it is acceptable to win the lottery out of sheer luck on page 790.

Third: Get more conflict out of each stroke of luck.

Usually luck removes conflict, removes struggles, and flattens an otherwise multifaceted plot. To avoid this you can add more conflicts and struggles for every lucky moment. Build them out of the envy of less fortunate characters. Make sure that the MC know that their luck is someone else's misfortune, and make them sorry for it. For instance, if you want to make them rich by winning the lottery, then make them alone.

I really like the answer @NofP gave.

I can add one more condition that would make it acceptable to the reader:

Forth: The character did something in the past that directly made the luck possible or probable.

Maybe the character helped someone early in the story who is responsible for the "stroke of luck." The important thing is that the MC didn't help the person expecting something in return. They helped that person because they thought it to be the right thing to do. Also, instead of a direct result, it is possible that the character did something that started a chain of events that, eventually, resulted in the character's "lucky break."