The Scotland international converted a brilliant spot-kick in first-half stoppage time to set the Parkhead outfit on their way to their first win of their Champions League campaign.
Regular penalty taker Kris Commons missed the 2-1 win with a hamstring injury but Forrest is keen to grab the role for himself after boasting to boss Neil Lennon about his flawless record from 12 yards.
The 22-year-old said: "I keep telling the gaffer that I've never missed a penalty. That's only the second one I've taken for the first team and I'm happy to take them."
Visiting stopper Jasper Cillessen did his best to put Forrest off his task as he went walkabout in his penalty area in the seconds before the Celtic forward stepped up.
But the ploy did not work as Forrest kept his cool to sweep home a fine finish.
"The goalkeeper was trying to play some mind games but I just focused on where I was going to put the ball and I'm just glad it went in," he added.
Beram Kayal added a second with the aid of a deflection eight minutes after the break before Lasse Schone gave the Hoops cause for concern in stoppage time with a late consolation effort for the Dutch champions.
Celtic now travel to the Amsterdam ArenA for the return encounter with Frank de Boer's side on November 6, hoping to keep tabs on AC Milan.
The Italian giants' 1-1 draw with Barcelona on Tuesday leaves them two points ahead of Lennon's men with three games remaining.
"We needed the three points so it was great to get it against a side like Ajax," said Forrest as he targeted a place in the last 16.
"The win was a huge boost for us. We had players out unavailable but the players that came into the team did very well and now we are looking forward to playing over there in Amsterdam.
"We had guys missing tonight like Kris and Scott Brown but if you look at out squad, we have strength in depth.
"If you take someone out, someone else will come in and do the same kind of job. Guys like Beram proved that on Tuesday night.
"It was crucial that we got the second goal early on. You don't want to be going into the last 15 minutes at 1-0 because it can get nervy. But with the 2-0 lead it gave us a bit of comfort.
"We are just taking things one game at a time now. We will look forward to Amsterdam and take it from there. But we are confident of winning over there.
"Hopefully we will have a couple of players back as well and be in stronger shape to get the win over in Holland."