Oli Shaw put the visitors ahead from a hotly disputed penalty on the hour mark after referee Nick Walsh penalised Richard Foster for handball but a Ross Docherty 20-yard strike salvaged a point for Thistle with three minutes left on the clock.
Partick boss Ian McCall described the penalty award as a "shocker", telling BBC Scotland: "I don't know what Richard is meant to do. His hand is not in an unnatural position. It's at the side of his body.
"You score late and you take a draw, there's relief there, but I thought we deserved to win the game."
Derek McInnes was disappointed not to make it two wins from two since taking charge of the Ayrshire outfit.
"Overall on the performance we're glad to come away with a point, but when you're in a winning position with so little time left you're disappointed not to take all three.
"It's important that when we get into those winning positions from here on in that we get all three."
The result means Arbroath remain a point clear of Killie at the top of the standings ahead of their trip to Raith Rovers on Saturday.
Elsewhere, bottom of the table Dunfermline will be desperate to bounce back from their 5-0 thrashing at Morton last weekend when they entertain Hamilton at East End Park.
Ayr go hunting for a third straight win as Morton visit Somerset Park while title hopeful Inverness are looking to end a run of three matches without a victory at home to strugglers Queen of the South.