Webster is referring to executive patronage, which allows an elected individual (such as the president of the United States) to appoint other individuals to government offices (rather than having those offices filled via election).
Patronage is one of the principles of Jacksonian Democracy, which Webster opposed during his career as a politician. At the time, and as it is now, patronage was regarded as a spoils system in which the winning political party was rewarded for their victory.
It seems to me that Webster could have easily left out "and patronage" from the quote provided without altering the meaning, but Webster is mostly remembered for his brilliant speeches as an orator and was likely attracted to the alliteration of "power and patronage".
It should also be noted that Webster was born in 1782 (five years before the Constitution was authored), and would likely have written or spoken that quote much later than the 1700s, most likely the 1830s or 40s.