Riders are using their mount's movement thus they do not provoke OAs

According to the rule for opportunity attacks:

You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

When mounts move, they use their own movement not that of the rider. Thus, when using a mount to move around, the rider is not using their movement, action, or reaction and thus cannot provoke opportunity attacks.

The only reason a rider can even ever be hit by an OA while using their mount's movement is because the mounted combat rules specifically allow for this:

[...] if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.

This rule specifically allows for an OA to be taken against a rider if the mount provokes, but does not otherwise change the rules for provoking. The rider doesn't provoke an OA even in this case, they can are only allowed to be targeted by the OA.

A rider never provokes OAs while using mounted movement. If a mount disengages, they cannot provoke OAs and neither does the rider. Thus no OAs can be taken against either.

Jeremy Crawford also agrees with this reading in a Tweet:

Q: if a mount takes the Disengage action does the rider provoke opportunity attacks? [...]

Crawford: No, since the mount isn't provoking them and the rider is being moved by someone else's movement (PH, 195).

Your example might not be possible depending on how your DM handles mounted combat turns

[C]an I ride up to an enemy, make a melee attack, then have my mount disengage and ride away, without provoking an opportunity attack?

There is quite a bit of debate on whether the rules say that a mount can act during a character's turn or not. I'll not side with one or the other here as this has been debated in several other questions.¹

If your DM rules that mounts must act on their own turn before/after the PC's then the example you give is not possible because the mount would have to move and Disengage then the PC would use their turn to Attack. But since it would no longer be the mount's turn, the mount could no longer move the PC away and the mount and PC would remain in front of the enemy.

The one way it is possible is by having the PC, on their turn, use their action to Ready an attack when their mount brings them in range of an enemy. Then on the mount's turn, it moves, disengages, and moves back, and the player uses their reaction to make one attack while in range. However, the PC would only get 1 single attack so it is not really perfectly the same as what you envisioned.

If your DM rules that a mount's movement and actions can be interspersed with the PC's² then this strategy is possible exactly as you say.

^{1 - See Does a controlled mount share its rider's turn? and Can a rider move using his mount, attack from his mount, then have the mount dash away? for further discussion. Note this is an example of designer rulingd seeming to confuse, not clarify the issue at hand.}

^{2 - Even for those DM's that agree that the rules strictly say that this cannot happen, this is a very common houserule (including at my table).}