If your total tax withholding (from your paychecks; not including estimated taxes) reaches 90% of your tax liability for this year, or 100% (110% for high earners) of your tax liability for last year, you do not have to pay estimated taxes and you will not have an underpayment penalty. (This is for federal tax; the rules for state underpayment penalties may be different.) If this year's tax is significantly higher than last year's, it may be that the 100%/110% of last year's tax level is lower than 90% of this year's tax. So first, check to see whether your withholding reaches this level (or whether it would be possible to increase your withholding in the paychecks remaining in this year to reach that level by the end of the year); if it does, you do not need to worry about estimated taxes.
If your withholding does not reach that level, you have to pay estimated taxes to reach that level, in order to avoid the penalty. The penalty is calculated on Form 2210; you have paid enough when the penalty calculated is 0. There are two ways of calculating the estimated taxes you have to pay each quarter: 1) you need to pay 1/4 of the estimated taxes you need to pay for the whole year each quarter, or 2) the Annualized Income Installment Method, which calculates the amount of estimated taxes you need each quarter based on your actual income received up to that quarter. The annualized income method is precisely for situations like yours where you suddenly received more income in later quarters, so you are not penalized for not having paid estimated taxes in earlier quarters when you didn't expect to need to pay it based on your rate of income then.
Assuming that your withholding was at a reasonable level for the income you were receiving in the first 3 quarters, you should not have a penalty for the first 3 quarters under the annualized income method. So you just have to pay 4th quarter estimated taxes (due January 15) so that your withholding and estimated taxes combined reaches 90% of this year's tax or 100%/110% of last year's tax, whichever is less. Yes, you can pay estimated taxes using the link you mentioned, instead of mailing in Form 1040-ES.