You now have two failures AND a success
On most rolls a result of "success" and a result of "failure" would be mutually exclusive and the roll would have to be resolved as one or the other. However, this is not true for a death saving throw (emphasis mine):
A success or failure has no effect by itself. On your third success, you become stable (see below). On your third failure, you die. The successes and failures don't need to be consecutive; keep track of both until you collect three of a kind.
The scenario you posit, in which the player rolled a natural 1 but had modifiers that made the result of the roll a 10+, concludes in a total of two failures and a success. The natural 1 generates two failures, and the result of 10+ on the roll generates one success, and these do not contradict one another. The particular rule for death saves of "keeping track of both" means that both of these results count.
Depending on your prior conditions, you may be able to simply apply both.
Prior: 0 failures, 0 successes. Now: 2 failures, 1 success.
Prior: 0 failures, 1 success. Now: 2 failures, 2 successes.
On the other hand, prior conditions may be such that the order in which you apply them appears to matter. However, while the process is different, the result is the same, and so it is still no problem.
Prior: 1 failure, 0 successes.
Either, apply success first, then two failures - result: you die and success is removed.
Or, apply failures first - result: you die and success is not applied.
A similar method of two potentially different processes but the same end result occur for other prior conditions:
1 failure, 1 success. Result: you die.
2 failures, 0 successes. Result: you die.
0 failures, 2 successes. Result: you are stable at 0hp.
2 failures, 1 success. Result: you die.
However, the more difficult cases are when the order in which you apply the successes and failures does matter, because applying one first will immediately result in either death or stability and having to ignore the other effect. This is the case when there are already 2 successes and either 1 or 2 failures.
Here, the DM has to decide by fiat which to apply first.
There is an optional rule for resolving these competing simultaneous effects in Xanathar's Guide to Everything (p. 77):
In rare cases, effects can happen at the same time, especially at the
start or end of a creature's turn. If two or more things happen at the
same time on a character or monster's turn, the person at the game
table - whether player or DM - who controls that creature decides the
order in which those things happen.
Since it will be the turn of the player actually making the death saves, that player then gets to decide the order in which to apply the results of "two failures" and "one success", which is obviously a considerable advantage in this very specific case.