The first step to winning any melee fight is knowing when to swing your sabre and when not to at all. Without going into detail, generally it is a bad idea to charge in the following cases:

1: You are charging uphill
2: You are fighting elite units or other units with high melee stats
3: Your troops are tired or wavering
4: You are surrounded

These types of situations are likely to end in lots of casualties and you losing the engagement.
If these are true for your opponent you have a melee advantage.

(You can check the melee stats of your infantry by right clicking on their icons. Trajan has a good guide where you can check the stats of all units in the game.)

After doing some testing, I've determined that a line infantry unit is likely to beat another line infantry unit if the defence of the defender minus the attack of the attacker is at most 1.

(defence - attack <= 1)

Units with slightly better melee stats than the generic line infantry include British Foot, Ottoman line (Nizam-i-Cedit), Polish Legion & Young Guard, and all grenadier units.

Units that are very likely to succeed in melee combat include: British Highlanders, Ottoman Janissaries, Hungarian Fusiliers (Austria), all foot guards.

All Russian infantry units receive a charge bonus, however the effect of charge bonus can be very subtle and I would hesitate to charge across flat ground. Beware though as Russian infantry charging downhill will slaughter anything in its path.

If you see a unit of foot guards that are very tired/exhausted and are fighting uphill with 0 supporting units, then a single unit of line will break them.

Although this guide emphasizes line infantry, cavalry can still be very useful in deciding engagements. Keep in mind, light cavalry units such as hussars generally cannot rout fresh line infantry charging head on. One strategy with light cav though is to pin down a unit of infantry with your own & swing around with hussars to charge from behind. The morale break from being sandwiched is likely to cause it to rout.

Only heavy cav units such as dragoons have the melee attack & defense to charge a line infantry unit head on and break it.

If these tactics worked out for you, feel free to leave a comment & describe what happened. If it doesn't work out, maybe you can start a discussion.