Veteran forward Hayter scored either side of half-time to help the Somerset outfit reach the fourth round of the FA Cup for the first time in nine years, with Moore adding the finishing touch in second half stoppage- time.
Former Bournemouth striker Hayter had netted only once this season prior to the cup clash with the O's and Johnson was elated with his striker's performance.
Johnson said: "He looked dangerous and sharp.
"What we asked for at half-time was a couple of good balls into the box and it was a good ball from Liam Davis, who I thought played very well today, and a good finish from Joel Grant and then Kevin Dawson hit a great cross which Hayter got on the end of.
"I did not want to take him off especially when you're on a hat-trick because you don't get too many chances over the years but I wanted to give Kieffer Moore a confidence boost, and Adam Morgan.
"Morgan nearly scored with his first touch and Kieffer did score when he was put in so there is a lot to come from them - they are only young lads so we need to bring them out of their shells quickly.
"Kieffer is a good finisher and a strong guy. In training when he finishes all our keepers say he has one of the hardest strikes with both feet in the club.
"He can be awkward sometimes, but I have had some big players and funnily enough he is not too awkward and has a decent touch.
"Give him time, because he is a fine specimen and hopefully we will bring him through."
Leyton Orient manager Russell Slade was not left too dejected by the cup exit, with league matters taking priority.
Slade said: "It was an unfortunate mistake from Jake in goal for the first and we rallied around at half-time and it was only 1-0 but the second was a bit of a killer really.
"The last one you can ignore because he is several yards offside but whether it is 3-0 or 4-0 they deserved to go through on the day.
"It is a difficult time for us because we have injuries and we are trying to bring one or two players in and still get results, so put this in a box and put it to one side.
"We're out of the FA Cup now so we can concentrate on the league and a very big game which we have on Tuesday."