It was a fine reward for the battling Bucks, who will be aiming for a third successive upset after Saturday's glory day at York City.
They pulled off a second upset in as many rounds as they followed up their home fourth qualifying round replay win against Kettering Town with the 1-0 win at York, Diego De Girolamo the hero with his 85th minute winner as new boss Steve Cunningham marked his first game in charge with victory.
This is the first time Buxton have made round two since 1958/59 and only the third time in their history, having lost a round three tie at Doncaster Rovers in 1951/52 in their best ever run.
Struggling Morecambe progressed with a 1-0 home win over League Two Newport County on Saturday – their first win in eight games in all competitions.
Buxton beef up with new quartet - and more could be on the way
Buxton manager Jamie Vermiglio happy with opening day point at Peterborough Sports
Buxton are held by 10-men Peterborough Sports in historic first game at National League North level
Commonwealth Games glory for Buxton swim ace Abbie Wood with four medals in Birmingham
Buxton pick up a draw at Peterborough Sports as both clubs debut in National League North
The Shrimps are only out of the League One relegation zone on goal difference, having only won four of their 16 EFL games to date, which will give Buxton huge encouragement.
Ties in the second round will take place across the weekend of Friday, 3rd December until Monday, 6th December, with the 20 winning clubs collecting £34,000 each from the competition prize fund.