There are no clear statements in Lord of the Rings as to the 'allegiance' of the Balrog, and so it is often wondered if perhaps the Balrog of Moria was acting independently or awaiting the return of Morgoth rather than acting as a servant of Sauron. As usual there are no direct statements to resolve this debate (it wouldn't be a debate if there were), but a careful examination of the texts can provide some interesting insights.
In earlier drafts of LotR there are indications that the Balrog might have been sent by Sauron;
"'A Balrog!' said Keleborn. 'Not since the Elder Days have I heard that a Balrog was loose upon the world. Some we have thought are perhaps hidden in Mordor [?or] near the Mountain of Fire, but naught has been seen of them since the Great Battle and the fall of Thangorodrim. I doubt much if this Balrog has lain hid in the Misty Mountains - and I fear rather that he was sent by Sauron from Orodruin, the Mountain of Fire.'"
Variants of this idea, some stating it more directly, can be found in note 11 of the same chapter;
"11 Parts of the underlying pencilled text of this passage can be made out, and the purport of Keleborn's words was very much the same - except that it was Keleborn (Galdaran) himself, not Galadriel, who raised a doubt:
'A Balrog,' said [Aran >] Galdaran. 'Of them I have not heard since the Elder Days ... had hidden in Mordor but of them naught has been seen since the fall of Thangorodrim. I doubt much if this Balrog has ... and I fear rather ... Orodruin in Mordor by Sauron. Yet who knows what lies hid at the roots of the ancient hills...' At the bottom of the page is a variant, added to the revised text but belonging to the same time, in which it is Galadriel who expresses the opinion previously given to Keleborn, and more decisively:
'No Balrog has lain hid in the Misty Mountains since the fall of Thangorodrim,' said Galadriel. 'If truly one was there, as is told, then it is come from Orodruin, the Mountain of Fire, and was sent by the Lord whom we do not name in this land.'"
While this version of the Moria Balrog's history was clearly rejected when Tolkien decided the Balrog should be the force which drove the Dwarves from Moria, it certainly demonstrates that Tolkien considered it possible for Balrogs to follow Sauron's orders. At that, there is at least one text mentioning such a situation;
"But at length after the fall of Fingolfin, which is told hereafter, Sauron came against Orodreth, the warden of the tower, with a host of Balrogs."
LRoW, Quenta Silmarillion - Ch. 11 ~143
It might be claimed that these Balrogs followed Sauron only because it was Morgoth's will. However, it is elsewhere indicated that Sauron commanded Angband during Melkor's long imprisonment, and that there were Balrogs in residence there. They might have been accepting Sauron's command only while awaiting Morgoth's return, but the same could be assumed of a Third Age Balrog.
There is also one passage which might suggest that the Balrog's appearance in Moria was partially due to Sauron's return;
"The power of Sauron, servant of Morgoth, was then again growing in the world, though the Shadow in the Forest that looked towards Moria was not yet known for what it was. All evil things were stirring. The Dwarves delved deep at that time, seeking beneath Barazinbar for mithril, the metal beyond price that was becoming yearly ever harder to win. Thus they roused from sleep a thing of terror that, flying from Thangorodrim, had lain hidden at the foundations of the earth since the coming of the Host of the West: a Balrog of Morgoth."
RotK, Appendix A.III - Durin's Folk
While this text indicates that the Dwarves roused the Balrog it also says that the 'Shadow in the Forest', aka Sauron, was looking towards Moria and that 'all evil things were stirring'. Following these statements immediately with the arrival of the Balrog certainly suggests that Sauron had some part in that event though it is possible the Balrog was merely stirred to greater activity by some sense of 'evil on the rise'
Set large against these arguments are;
"Long ago they fell under the dominion of the One, and they became Ringwraiths, shadows under his great Shadow, his most terrible servants."
FotR, The Shadow of the Past
"At length he [Sauron] resolved that no others would serve him in this case but his mightiest servants, the Ringwraiths..."
UT, The Hunt for the Ring
If the Nazgul were Sauron's most terrible servants (as is stated a number of times) then it is natural to conclude that the Balrog did not serve Sauron. Unfortunately, this status of the Ringwraiths dates back to very early drafts of the story, and co-existed with the passages quoted above from Treason of Isengard where the Balrog apparently was a servant of Sauron;
"'But all the Nine Rings of Men have gone back to Sauron, and borne with them their possessors, kings, warriors, and wizards of old, who became Ring-wraiths and served the maker, and were his most terrible servants."
RotS, Ancient History
It might then be supposed that when Tolkien considered Sauron in command of Balrogs at Orodruin he was referring to the earlier 'less powerful' sort of Balrog. And yet, even those early Balrogs were amongst the most powerful of Melkor's creations;
"Now those drakes and worms are the evillest creatures that Melko has made, and the most uncouth, yet of all are they the most powerful, save it be the Balrogs only."
BoLT II, Turambar and the Foaloke
Based on this it seems clear that either the Nazgul were vastly more powerful than commonly supposed (the equal of dragons, which seems unlikely), the Balrog was not a servant of Sauron, or Tolkien was not considering the Balrog when he called the Nazgul Sauron's greatest servants.
One further possibility might be that the Balrog was allied with Sauron rather than serving him, and yet there is some reason to doubt even this. In TT, 'The Uruk-hai' there are three groups of Orcs. Saruman's troops under Ugluk, Sauron's troops under Grishnakh, and Northerners "...come all the way from the Mines to kill, and avenge our folk." It seems clear that this third group was from Moria (the Mines). Unlike the other groups they do not indicate that they have received orders. Nor do they side with Grishnakh's orders that the prisoners should be taken to Lugburz. Instead, one says; "I wish to kill, and then go back north.". If the Balrog WERE under Sauron's orders or allied with him we might reasonably expect the Orcs cohabiting with him to be so as well - yet there is no indication here that they were. They have no interest in the Fellowship except to kill them. Likewise, in the appendices it is stated that Lorien was attacked three times - from Dol Guldur. If Sauron were allied with the forces in Moria they would reasonably be expected to join in coordination with these attacks, yet there is no indication that they did so.
Thus, while there seems to be ample evidence in older texts that Sauron COULD control Balrogs there is little in Lord of the Rings itself to suggest that he DID command the one in Moria.