I think you did fine. Lantus acting normally won't cause sudden lows, it doesn't start working for 1-2 hours and won't reach a peak for 6 hours. But I said "normally." Lantus must be injected into the subcutaneous fat layer and you must not inject it into muscle or veins. So the proper technique is to choose an area of your body with a good layer of fat (usually your abdomen) and pinch up to inject. This is the technique that you should have been taught, but maybe not. If you misinjected Lantus, it is possible that it can cause a sudden low. There are numerous posts about injection technique, I started one here.
In the meantime, if you are on a single Lantus injection a day you may actually find it safer and less stressful to simply take it in the morning. That way any problems you have can be handled while you are awake.