Faced with near gale force winds and torrential rain, the Scarlets' defence held firm for 80 minutes, with all of Ulster's points coming from the boot of full back Clinton Schifcofske.
"Our defence was excellent again, which is hugely encouraging and a big positive for us," said Easterby.
"There was a huge amount of character shown by the boys because regardless of the conditions this is always a tough place to come and play.
"We have got to take this win now and build on it."
The victory means the Scarlets finish the first weekend of the Magners League season in second place behind the Ospreys. They have matches against The Dragons and Connacht in the next seven days and a chance to build some early-season momentum.
The skipper, however knows that nothing can be taken for granted, he said: "We have got a tough week, (Dragons away on Wednesday, Connacht home on Saturday), but it will be great to finish off with a couple more wins.
"We are not fancied by many people, but we are just keeping focused on our job.
"And if we can perform like we did tonight I am sure we can get more positive results like this one."