Town's joint top-scorer Pablo Counago had a rare off night in front of goal, passing up several presentable chances including a complete air-shot 12 yards out in the first period.
Ipswich had the better of the first half and appealed for a penalty when debutant Alan Quinn clashed with Peter Halmosi in the box, but referee Tony Bates said no and it looked the correct decision.
Plymouth keeper Luke McCormick saved easily from Counago and Quinn, while Tommy Miller had a firm shot saved as Ipswich pressed before Owen Garvan horribly miscued from a Counago square ball.
Danny Haynes blazed over after cutting in off the right wing, Jon Walters looped a header over from a David Wright cross and Garvan then grazed the post with a low 20 yarder after Haynes, Walter and Counago were all involved in the build up.
Counago hooked over from seven yards when a Garvan header found him in space from a corner, before a rare Plymouth moment of opportunity saw Alexander go down well to claim a Halmosi free-kick.
Counago missed the ball completely when Haynes seemingly put a goal on a plate for him and half ended with Walters freeing Counago, who drew the defender but then slid his shot just wide.
The second period was to prove more even, although early on it was all Ipswich with Alex Bruce seeing his header from Garvan's corner cleared off the line by Halmosi with Gary Sawyer then clearing Jason De Vos' follow up off the line by the opposite post.
Alexander saved well from Halmosi and then almost immediately created a goal when his long punt was diverted into the path of Counago, whose lob was clawed away for a corner by McCormick when the striker should really have scored.
The visitors had a great chance to go in front on 61 minutes when Dan Harding handled while trying to stop Paul Connolly reaching a loose ball, but Alexander dived to his left to parry MacLean's spot-kick - his third penalty save of the season.
McCormick made a good low save from a Walter's header after Wright's cross found him, but this was topped by Alexander's magnificent reflex grab to deny Rory Fallon when he came steaming in at the far post.
Miller shot narrowly wide for Ipswich before Argyle sub Jermaine Easter miscued his header from Chris Clark's fine cross when he should have done much better.
Ipswich's bad luck was summed up in the final few minutes as Counago headed against the post from another inch perfect Wright cross.
Then subs Gary Roberts and Alan Lee teamed up with Roberts crossing for Lee to head at goal, but Mathias Doumbe cleared off the line.