Paying by January 15 probably will not eliminate the penalty because you are still paying late. The tax was supposed to be paid over the course of the year. But if you pay by January 15 the penalty will be less than if you wait and pay when you file your tax return. If you can, you should pay the full amount by January 15, not just the last three months. If you can't pay the full amount, pay as much as you can.
If the amount that you owe is less than $1,000 there will be no penalty. But if your withholding was too low, I suspect you owe more than that.
