Ipswich started well and after just four minutes both Jon Stead and Pablo Counago had efforts cleared off the line by Romain Larrieu and Rory Fallon respectively following an Owen Garvan corner.
Town then had strong penalty appeals rejected by referee Darren Deadman as Marcel Seip clambered all over Counago.
Plymouth gradually came into the game and Richard Wright saved shots from Paul Gallagher and Simon Walton, while the latter also shot over.
Ipswich came closest to opening the scoring in the first half when Jon Walters crossed for Jon Stead to thump a fierce volley against the post.
The visitors started the second half on top and first Walton fired a free-kick over and Gallagher's angled drive was saved by Richard Wright.
Plymouth then hit the woodwork through Fallon's 12 yard shot from Gallagher's cross, with Walton tamely heading the rebound straight to Richard Wright.
Walters volleyed over for Town when Seip and Chris Barker got in each other's way, before Chris Clark volleyed over from Karl Duguid's cross.
As time ticked on, Ipswich pressed more and more with Larrieu saving from Stead and David McNamee diverting away a goalbound Walters drive.
Danny Haynes scuffed a shot wide late on and Larrieu saved from Alex Bruce and sub Kevin Lisbie, while Plymouth thought they had won it with a minute to go when Jamie Mackie netted but he was flagged offside.