Blackwell was named manager of the month for March on Thursday after a six-game unbeaten spell, and now a third successive league win has left second-placed Birmingham, who clash with leaders Wolves at St Andrews on Monday, looking anxiously over their shoulders.
Greg Halford, on a season-long loan from Sunderland, gave the Blades a 19th-minute lead with a raking 25-yard drive - his seventh goal of the season - and Darius Henderson headed home Kyle Naughton's measured cross eight minutes later.
Blackwell, whose side had the game wrapped up in the first period and cruised through the second half against Jim Magilton's team, said: "I thought in the first 45 minutes we were unplayable.
"I said to the boys (at half-time), 'Let's be as professional as we can and finish the game off', and in the second half I knew Jim and Ipswich had to go for it because it was maybe their last chance to get in the play-offs.
"When we had to defend we did so very well, but we still had two or three great chances to finish it off, so all in all we're very pleased."
United moved level on points with third-placed Reading, who have a better goal difference, and appear to be the form team ahead of a six-game run-in.
They have lost just two of their last 19 league matches.
Blackwell added: "All we can do is keep putting pressure on them. I've been saying for quite a while now that it would be disrespectful and unprofessional if we don't give it a go.
"We're trying to catch them and put them under as much pressure as we can and if they trip up, hopefully we'll be able to take advantage of it.
"But it was a big three points today."
Ipswich manager Magilton conceded his side's chances of making the play-offs looked remote after starting the day seven points adrift of the top six.
The Irishman said: "I very much doubt it now, but we'll certainly keep going.
"We're going to try to win as many games as we possibly can, but we obviously have to play a lot better than that.
"I won't allow a stroll towards a lovely end of the season. The players will be on their toes right to the finish."
Magilton was at a loss to explain why his players were made to look lightweight, admitting they "lacked punch" and had been "disheartening" away from home all season.
He added: "There was no indication going into the game. We looked ready, we looked sharp, I just think the first goal was very important.
"At half-time we were very positive and knew the next goal was crucial, but we huffed and puffed and Sheffield United shut the game down and defended well."