Batt, a summer signing from non-league Fisher Athletic, took just 64 seconds to break the deadlock.
Chris Whelpdale doubled Peterborough's advantage in the 43rd minute as Darren Ferguson's classy side extended their unbeaten run in the league to six games with a third straight win.
Walsall had gone into the game on a good run of their own - four wins in five matches - but they never got over falling behind to Batt's early goal.
Substitute Troy Deeney struck deep into stoppage time when he headed home a corner from Chris Palmer to grab a consolation goal.
But Jimmy Mullen's team were fortunate that they did not slip to a heavier defeat as they struggled to keep the pacey Posh attack in check.
The hard-working George Boyd was particularly unlucky not to get his name on the score sheet.
He saw a 56th minute shot blocked wide and then had his goalbound drive deflected on to the post 11 minutes later, while Craig Mackail-Smith also hit the woodwork.
Walsall toiled to find a way back into the game but Peterborough goalkeeper Joe Lewis pulled off stunning saves to deny Jabo Ibehre and Ishmael Demontagnac, before he was beaten by Deeney's late strike.
Batt's strike was unfortunate for Walsall goalkeeper Rene Gilmartin, who was making his first league appearance for more than two and a half years.
When Mackail-Smith delivered a left wing cross, Batt saw his low shot saved by Gilmartin.
But the Republic of Ireland under-21 international keeper was still on the ground when Batt headed the rebound home.
While Peterborough had to wait until the stroke of half-time to double their advantage, it was a goal that had been coming.
Walsall had briefly threatened when Gerrard and Ibehre hit the bar with headers in the 23rd minute, but Peterborough were always a threat.
Gilmartin saved from Mackail-Smith and Batt was just off target after good work by Russell Martin, before Whelpdale made Peterborough's pressure pay two minutes before the interval.
Paul Coutts rolled the ball into his path and he beat Gilmartin from 20 yards.