Timlin scored his first goal since March 2016 in the 4-1 League One win, which saw Southend move up to fifth.
"All the boys said before the game it was almost written in the stars, it was going to happen," the 31-year-old said.
"It's a fantastic feeling, the birth of my first child, Rafe, and getting the win tonight and scoring."
He added to BBC Essex: "We knew how important this win was for us because if Peterborough beat us we would be level on points, and with Millwall drawing at home it's a massive result."
According to Timlin, his son's birth came around an hour after team-mate Anton Ferdinand's own baby girl arrived. The Southend captain missed Tuesday's game at the ABAX Stadium.
Timlin said he only needed to be asked once by manager Phil Brown whether or not he wanted to play the match against fellow play-off hopefuls Posh.
"My missus started having some tweaks on Friday night and I was worried about the game on Saturday," he added.
"Then I was hoping it was going to happen on Sunday, then when it didn't happen I didn't want it to go close to Tuesday's game because we're on such a good run, I wouldn't have been happy to miss the game.
"Obviously you want to be there for the birth of your child, my first one, but my missus knew the situation - if we had the baby on time I definitely wanted to be playing."
Southend United midfielder Michael Timlin said he was desperate to play against Peterborough on Tuesday, one day after the birth of his first child.