Daryl Murphy was the match winner with a 38th-minute tap-in following good work from Paul Taylor and Frank Nouble.
Ipswich are now four points off sixth place but McCarthy insists others are better placed to claim a play-off spot.
The Town boss said: "I think the top five are clear and Wigan are the favourites for the sixth spot.
"They are ahead of us and they still have players from the Premier League last season. But we are still in there, scrapping away."
Of his side's single-goal victory, Town's fourth consecutive clean sheet at Portman Road, McCarthy added: "I think it was the right result - just. But I don't think we dominated Birmingham by any stretch of the imagination."
After a poor 35 minutes, the match turned on two big chances.
Murphy gave his side the lead after tapping home Nouble's low cross following great work from man-of-the-match Taylor.
The goal gave Ipswich a new lease of life and they deserved the three points after an impressive second-half showing.
But Birmingham should have restored parity on 63 minutes when Lee Novak was presented with an identical chance to Murphy's opener.
Sub Chris Burke did well to beat Tommy Smith and his low centre should have brought the equaliser, only for Nowak to fire over when it looked easier to score.
Anything but a victory would have been harsh on Ipswich who improved after taking the lead.
Taylor - who only started after new loan signing Jonny Williams got injured in training on Friday - was at the heart of all their best moments and he twice went close to doubling the lead before Birmingham keeper Darren Randolph saved from Paul Anderson after a woeful back pass from Jonathan Spector.
Birmingham boss Lee Clark had little complaints in his post-match summing-up.
Clark said: "We were a little toothless. We didn't have enough quality. The defence was solid and we stood up to a big physical team in Ipswich - we did the ugly stuff well.
"But we didn't do enough when we had possession. We lacked that quality and I don't think we deserved anything from the game."
And Clark felt Novak's gilt-edged chance was typical of his side's overall performance.
He added: "Lee Novak usually puts those chances away but unfortunately he hit it over the bar.
"A chance like that, away from home when you are not creating many, has to be taken and that seemed to be a problem for us today."