The 19-year-old English-born of Ghanaian descent came on for Scott Arfield in additional time, with his manager insisting he deserves more playing time henceforth.
"I was waiting to get him on. We would have maybe got him on for a few minutes more, not for any gimmick, but because he deserves the right," Dyche told his club's website.
"He has been training with the group and training really hard and showing us signs and glimpses of what he's about.
"He's a young man and to make his Premier League debut at Anfield is not a bad start," he added.
Agyei hopes to build on his successful debut to achieve success in the future.
"It was a great experience for me to make my debut. It's just disappointing not to get a result," Agyei told Clarets Player HD.
"It's a massive step forward and hopefully I can kick on. I just need to keep training hard, try and get as many minutes as possible and hopefully score some goals," he added.
Burnley will travel to the Stadium of Light on Saturday to lock horns with relegation-threatened Sunderland.
Goal understands that Ghana FA has opened talks with the striker's parents to convince him make a nationality switch.