Paterson levelled with five minutes remaining at London Road to move the Clarets to within four points of the Championship play-off places.
And only a terrific stoppage-time save from Posh keeper Bobby Olejnik denied Paterson a dramatic winner as his glancing header from point-blank range was clawed to safety.
Burnley had earlier hit the front after just 65 seconds through fit-again top-scorer Charlie Austin, but they were staring at a shock defeat after two goals in the space of five minutes from Posh skipper Tommy Rowe midway through the second period.
Paterson's leveller ensured a share of the spoils that did neither side any favours.
Dyche said: "It's a pleasing point in one respect as it takes us closer to the play-off places, but it's also a frustrating result as we felt this was a game we could have won.
"We enjoyed a fantastic start and controlled the first half of the game on a very awkward surface.
"Peterborough had a real go in the second half, as you would expect from a team fighting for their lives at the bottom of the table.
"We conceded a poor goal to be pegged back and then found ourselves behind, but our boys always keep going right to the bitter end.
"It's always pleasing to see strikers scoring goals and two of them got on the scoresheet. We're four points off the top six and in good shape but we're hungry for more than that."
- Sports & Recreation
- Charlie Austin