The England international beat former Aberdeen goalkeeper Bobby Clark's 43-year-old record of 1,155 minutes without losing a league goal as Celtic beat Hearts 2-0 at Tynecastle.
The record now stands at 1,215 minutes after Forster's 13th consecutive Scottish Premiership shut-out.
And the 25-year-old now has his sights set on building on his England debut, which came in a 2-0 defeat by Chile in November, by securing his place in Roy Hodgson's squad for Brazil.
"You can just do what you can do," Forster said. "You can try your hardest each week and just do the basics well. All you can do is concentrate on yourself and doing the best you can for Celtic. You just have got to hope that's enough going into the summer."
Forster has played down the potential record leading up the game and will continue to focus quietly on his aims with Celtic goalkeeping coach Stevie Woods.
"I have my core group and they have all been very positive and encouraging," he said.
"Me and Woodsy sat down at the start of the season and came up with goals of what we wanted for the season. This obviously wasn't one of them but it's been a real bonus to go on this run."
Celtic manager Neil Lennon was delighted his side set the new record after withstanding some aerial pressure from Hearts in the second half.
Leigh Griffiths fired them into a 57th-minute lead with a well-taken finish on the counter-attack - his first goal since signing from Wolves in January - and Teemu Pukki added a second on the break with the last kick of the ball.
Lennon, whose team remain unbeaten in the league after their 15th consecutive Scottish Premiership win, said: "I sit here a very proud manager.
"We have just broken a long-standing record, and it's long-standing record because it's a very difficult thing to do. It's a great feather in the cap for the club, and obviously we maintain our unbeaten record in the league.
"I thought we played with great composure. We knew it was going to be a difficult game against a big, physical team. The goal to break the deadlock was fantastic and that will do Leigh the world of good.
"It's a big step up in his career and you saw the relief and what it meant to him to get the goal. We are hoping that's plenty more to come from him.
"He has that accuracy with both feet, he hits the ball with great control and hits it so hard, it's difficult for keepers to get down that quickly. He seems to find the corner of the net with great regularity. His goalscoring record is very good, and that's the reason we brought him in."
Former Hibernian striker Griffiths celebrated excitedly in front of the Hearts fans before sprinting to embrace his manager.
"I think it was more relief than anything else," Lennon said.
"Obviously there was an added incentive for a Hibs fan scoring against Hearts, but that's not the main focus now. He has broken his duck for Celtic and it's a brilliant start for him."