You already bolded the answer.
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
If you're not taking the Attack action on your turn, then you can't use Extra Attack. Taking the Attack action doesn't just mean attacking - it's a specific action you can perform, listed under Actions in Combat.
Because the feat doesn't specify a clear timing, your attack happens after the enemy's
Page 252 of the DMG has advice about Adjudicating Reaction Times. Specifically, it suggests following the timing specified by the reaction:
For example, the opportunity attack and shield spell are clear about the fact that they can interrupt their triggers. If a reaction has no timing specified, or the timing is unclear, the reaction occurs after its trigger finishes, as in the Ready action.
The reaction from sentinel doesn't specify that the attack interrupts the opponent, or that your attack occurs before their attack hits or misses. At best, this is unclear timing and at worst there is no timing specified at all. So, based on the advice in the DMG, the attack given by the feat would occur after the attack from the enemy. So, killing the enemy with the reaction attack won't stop their attack; their attack has already happened.
Similarly, if you were to shove or trip somebody with a reaction like the one granted by the feat, the reaction would occur after the enemy attack, and so tripping them or knocking them away wouldn't stop their attack either.
Of course, the DM does have license to rule otherwise
As is typical of 5e, this situation really comes down to how the DM feels about the specifics of the timing. If the DM were to disagree with my assessment, and felt that the reaction description does actually specify that it occurs before the enemy attack can hit, then they would be justified in using this same reference to make the opposite conclusion that I have. So, take my analysis with a grain of salt and consult your DM for the ruling that will apply to your game.
Best Answer
After the triggering attack is resolved
From the DMG...
So, you have cases like the Shield spell where it says...
So in that case, it specifies that the Shield spell works on the attack that triggers it, so the Reaction to cast it must occur before the attack lands.
However, there is no such language in the Sentinel Feat--the timing is not specified, so the general rule of "reactions occur afters their trigger finishes" applies.