The question is vague. I am not criticizing the question. It's a good question. But as other answers have pointed out, while the definition of high salary is clear, the definition of better politician is not.
That being said, everything is vague. If we want to find samples where high salary improves quality of government services, I would look to Singapore.
Lee Kwan Yew raise salaries of top government officials.
Singapore, is well governed.
That is the speech.
The result works. The reasoning is also reasonable.
There is a quora question and answer that explain this very clearly https://www.quora.com/Why-is-the-salary-of-Singapores-Prime-Minister-so-high-Considering-U-S-President-Barack-Obamas-base-salary-is-400-000-and-considering-Obama-governs-a-country-that-is-more-than-50-times-more-populous-than-Singapore-isnt-a-1-7m-salary-absurd
Look at CEO's and managers' salary in US. You don't pay them peanuts. If you offer peanuts, those who come to work for you will be assholes.
By offering CEO's stock options and bonuses businesses properly align the CEO's interests with the businesses goal.
So it seems that the 2 important factors are
1. Large compensation to attract people that deserve it
2. Proper alignments between CEO's/politicians' income and the corporation/the state's interests.
The problem with politic is that unlike in corporation, the state's goal is not clear. Hence properly aligning politicians salary with interest of the state may be not clear to most "stakeholders" of the state, such as voters.
It is possible that politicians simply raise their salary because they want to be richer and not because it attracts better candidate.
In fact, why not go all the way? Why not turn prime ministers into emperors with palaces and harem. That should "attract" the best right?
It turns out, having emperors with ridiculously high salary and "benefits" leads to certain conflict of interests with the people. We got constant welfare for dynasties after dynasties in pretty much all over the world till democracy. If the pay is too high, everyone wants to get the top job.
Most countries pick the opposite strategies. They offer peanuts for their politicians. What happen is they got monkeys.
Why would guys, like Bill Gates, would want to work as governor in Indonesia. Not only that the salary is low, you can go to jail due to bigot mob. Needless to say corruption was rampant in Indonesia.
Is this evidence a slam dunk evidence? I do not know.
If I look at China, for example, I am quite confused my self.
Corruption is low. However, some governors are jailed for accepting $200k bribe, I've heard. And that does not make sense to me at all. Why would a governor that oversee trillions of dollars worth of economy get bribed by a mere $200k. Well, it turns out that the governor's salary is only $20k.
That being said, China is a reasonably well governed country despite low salary for its politburo.
I think the issue is politicians' "real income" is not ever really cheap. Suharto may have low salary but he makes billions of dollars due to various rent seeking.
I would say, salaries of politicians influence the quality of their services in complex ways. I would suggest that salary should be in line with salary in private sectors with reasonably good candidate.
Singaporean government seems to be unique. Their salary is about right. I read an article where someone quite their private sector job with $3 million salary with a $500k job in government sector. So we have it both ways it seems. We got a good guy "that can earn $3 million in private sector" and is obviously motivated reasonably by "devotion".