He used the same principle as Hipparchus and all other astronomers, including those more ancient than Hipparchus.
The idea is the following: if you can determine some fixed moment in the year (like a solstice or an equinox) with error E, then by doing this N years apart, you divide the number of days between two successive events on N, and your error will be E/N. So larger the N is better your accuracy is. For example, if you know the only the number of days from one equinox to another 1000 years later, your can determine the length of a year with accuracy 1/1000 of a day.
We don't know exactly what ancient observations Hipparchus used (but it is almost certain that he had access to some ancient Babylonian observations few hundred years before him. Hipparchus own observations are preserved by Ptolemy.
But Omar Khayyam had a much longer period at his disposal, at least from Hipparchus to his own time. (And more, since Ptolemy also preserved more ancient observations). Accordingly his accuracy was higher.
Speaking of the techniques of fixing the exact moment of a solstice or equinox,
there was no much improvement from the times of Hipparchus to the time of Khayyam.
So using my notation, their E was of the same order of magnitude, while Khayyam's N was much larger.
You can obtain even better accuracy by computing the time from the earliest observation known to you to an equinox which you can observe yourself.