No one is answering this question because 1) it is an over-simplification and 2) the unsimplified answer is really hard to find. BUT, I was really interested so I spent way, way too long looking into this:
First, to answer your question itself. Here are Dyson's career stats:
Here are Eloy's
You can see Eloy generally adds about -3% to -5% to success rate, whereas Dyson generally comes out to about 5% added in large sample sizes. So we are looking at probably a 10% swing in catch probability - Eloy has a success rate of 83% while Dyson has 93%. Of course, the thing this does not capture is that Eloy's biggest weakness may be going after balls in the gap - we have seen he is very slow to get the ball back in. So let's say a batter has, idk, a 10% chance of taking an extra base against Eloy that they would not have gotten against Dyson.
Eloy has played LF for 5733 opponent PAs, and he has fielded 542 balls in that time, so on average he fields a ball in 10% of the at-bats he plays in LF. Now, the chances may have been lower or higher in this inning, depending on batter profiles, pitcher arsenal and velocity, etc, but lets talk on average. Of the balls Eloy fields, about 60% are catching flyballs, the rest are retrieving a ball from somewhere in the outfield.
So each batter has an extra 1% chance of getting on, and an additional 1% chance of taking an extra base. Most of the balls that Dyson catches that Eloy won't would be pretty tough, so lets say each batter has a .25% better chance of a single, .65% better chance of a double, and a .1% better chance of a triple, since those are rare. Add in the effect of taking an extra base, almost all of which would be stretching a single to a double, and I'd say (very roughly) that the average batter has .25% better chance of a single, 1.5% better chance of a double, and .25% better chance of a triple.
Now to answer the question you are simplifying - how much does that affect the Sox overall chance of winning? And how does it compare to the offensive advantages of leaving Eloy in the lineup?
This is a pain to answer mathematically - you have to recursively calculate a ton of stuff because the win percentages change based on which batters get on (i.e. are there no outs still or two outs). You can use Markov Chains to calculate expected runs from 24 base-out states but that doesn't take into account the batters and the pitcher or handedness or a tons of other super important stuff. Over the last year I built a basic Monte Carlo baseball simulator to account for these things. It isn't particularly advanced, but it can give us a rough estimate here, so I'll take the opportunity to use it instead.
To answer the latter question first, let's look at how important Eloy is if you assume the Twins at least tied it up, and also don't take fielding into account. I ran 100,000 simulations, and I found that with Eloy in the lineup, the Twins won 60.5% of games when they scored at least 2 in the top of the ninth. With Dyson in, the Twins won 61.3% of those situations. Note that the Twins chances of winning here are much higher than 50% because in some of these situations they scored more than 2 runs and took a lead into the bottom of the ninth.
So if we assume the Twins at least tie it up, having Eloy in the lineup over Dyson is worth about 0.8% towards the White Sox' chances of winning. Not a ton, and the overall effect is that the Twins have a 6.11% chance of winning with Eloy in while 6.30% chance of winning with Dyson in. But again, that is ignoring fielding, so let's see what happens when we take into account the fact that Dyson lowers the chances of the Twins tying it up in the first place.
I ran another 100,000 simulations for both Eloy in and Dyson in, this time adjusting the opponent chances of hits as described above. With Eloy in the lineup, the Twins had a 6.18% chance of winning. With Dyson in the lineup, their chances of winning were 6.24%. In case you thought I woefully underestimated the importance of Dyson's defense, I reran the simulations with the defense having 5x more of an effect. In this case, the Twins win% with super Dyson was 6.19%
Basically we are talking near negligible differences, and the White Sox chances of winning were slightly better with Eloy in. Really though, this exercise just goes to show how tiny of an effect individual strategic choices usually have on a team's chances of winning. These things just don't matter all that much in baseball, and I'll argue about them until blue in the face anyway.
TL;DR: The difference between Eloy and Dyson on overall win percentage was negligible, and there is a mathematical argument that leaving Eloy in was the correct choice.
BTW, I saw someone mention that RR only left Eloy in because he was leading off the ninth. So I reran the simulations with the scenario that Eloy instead made the last out in the 8th, and EE was leading off the ninth. In this case, leaving Eloy in gives the Twins a 6.22% chance of winning, while putting Dyson in gives the Twins a 6.17% chance of winning, and now subbing Dyson is (again, barely) the correct answer. So it was fair to treat Eloy leading off as a major factor.
If you think I messed up any of the estimates badly let me know, I am happy to rerun with different settings.