させる
So, this is a new enough grammar point for me that I’m going to preface this that you should take it with a grain of salt. But my understanding sort of boils down to this.
We are looking at two grammar points mushed together.
させる + もらう.
させる can be for either permission (let someone do) or coercion (make someone do). It’s context dependent.
もらう is “have someone do something.”
Combined, I think they sorta take on a meaning wherein the speaker essentially gives themselves permission, from what I was reading earlier. I’ll edit this post when I find the link.
If i were to translate the sentence literally:
「この一局きっちり勝たせてもらいます」
“In this game, I will have you allow me to win!”
Which obviously is awkward in English. The spirit of the message would then become,
“I am going to win this game, no matter what!” …sorta.
I could be off-base, but that is what I got from it.
Editing to add: As @2OC3aOdKgwSGlxfz mentioned above, it’s also polite speech, so that interpretation is probably a bit on the rougher side. “I’ll take the liberty of winning, if you’ll allow it.” may be better? It’s difficult to map onto English.
Edit 2: After reading further into that link (I stopped at the initial explanation of what the combo meant, earlier), they do point out that the politeness can often be used to mask a “no matter what” type of feel, so I’m happier with that explanation than it him being super polite, given the context, especially with 「きっちり」being in the statement.
Found the link: https://forum.wordreference.com/threads/させてもらう.3790593/