Referee David Webb did not give the goal initially, believing that John Brayford had cleared Andrews' header off the line, but after consulting with his assistant he eventually awarded the goal.
Ipswich had failed to keep a clean sheet in their last eight outings but recorded their sixth shut-out of the season thanks to a series of good saves from goalkeeper Arran Lee-Barrett.
He twice denied Jamie Ward and also saved from Craig Bryson in a busy opening 45 minutes.
It was the Rams' sixth defeat in their last eight games while the win helped move Ipswich five points clear of the npower Championship relegation places.
Lee-Barrett was thrust into the starting XI as loanee David Stockdale was recalled by Fulham following the injury to Mark Schwarzer, and he made several good saves in the first period.
He was tested in the 16th minute when he had to beat away Ward's drilled effort after the winger had cut in from the right flank.
At the other end, Andrews' long-range shot moved in the air and goalkeeper Frank Fielding had to push the ball away.
But it was his opposite number who continued to be kept busy, making a fine double save to preserve Ipswich's clean sheet in the 27th minute.
Bryson's initial shot was tipped away by the Town stopper and he then palmed Ward's effort over after the winger had done well to create some space.
But it was the hosts who went ahead just 29 seconds into the second half through midfielder Andrews.
The Republic of Ireland international scored twice in the first four minutes of the second half against Barnsley last weekend and he broke the deadlock this time with a header.
Andrews met Aaron Cresswell's cross with a header which Fielding flapped at and the ball went over the line before Brayford could hack it clear.
Referee Webb did not give the goal initially but consulted with his assistant after Ipswich protested and he eventually gave the goal to Andrews, his ninth of the season.
The Tractor Boys looked to extend their lead and Fielding had to divert Lee Martin's long-range try over the crossbar. From the resulting corner, Ibrahima Sonko found some space but he headed over.
Substitute Jimmy Bullard came close to scoring just seven minutes after coming on but his venomous volley cannoned off Jeff Hendrick's back.
The visitors scarcely threatened Lee-Barrett's goal in the second period and Ipswich were able to coast to victory after four successive defeats at home.