When Andrews completed his hat-trick in the 48th minute Blues were coasting as Ipswich had not even got on the scoresheet.
But somehow Blues conspired to allow Ipswich back into it and half-time substitute Zak Brown completed his treble of goals in the 83rd and 88th minutes to give the visitors the spoils.
Remarkably Blues opening and previous PDL game at Wast Hills ended 5-4 also, to Crewe Alexandra.
In the first-half Blues were vibrant and well worth their advantage. And but for more care, they could have been well out of sight.
In the 26th minute from a Miguel Fernandez pass, Oriol Soldevilla came in from the right and hoisted a gilt-edged chance over the crossbar from deep inside the area.
Then from a dangerous cross from the right by Nico Gordon, Dylan Crowe did ever so well to deflect Andrews' near post touch at goal off target.
Blues then hit Ipswich with a double goal burst in the 33rd and 37th minute - both from right wing short corners.
First, Fernandez came round onto the ball on the angle near the box and chipped it in for Andrews to swoop and power a header into the top corner.
Then Ipswich were undone again by a clever routine; this time Fernandez backheeled the ball to Ryan Stirk to clip over from the byline and in came Andrews again, with a diving header from close range.
With two minutes to the break, Adan George should have added another goal after riding through two tackles to get into the area but, with the goal at his mercy, he lashed hard at the ball and sent it well wide at a horrible angle.
Even then, in added-time, there could have been more: from a Gordon cross on the run that was headed clear, Fernandez chested the ball under control and popped it back towards the top corner with a deft lobbed volley, but Bert White made a fine fingertip save.
Three minutes after the interval Blues extended the lead. It came when George overran the ball when trying to drive through down the middle; he blasted it at a defender who came to tackle and it spun off wickedly at an angle to Andrews, to the right inside the box.
Andrews chopped cleverly onto his left foot and let fly with a low shot that took a slight deflection on its way just inside the far post.
That really should have been that - but no.
Ipswich got one back in the 52nd minute through Brett McGavin, who whipped in from the penalty spot after Mitchell Roberts was penalised for a soft handball.
It was 3-2 in the 62nd minute when Brown flew in to connect and head into the roof of the net following a burst down the right and a fine centre.
Then Ipswich struck again from the spot, via McGavin, two minutes later to level things up.
Aaron Clayton got a hand to the penalty, as it went to the left once more, but there was more power on this attempt and beat the 'keeper. The award of the spot-kick was even more contentious as Tate Campbell made a firm, but fair - clearing getting the ball - sliding tackle on McGavin as he tried to hit the byline.
After Clayton made a sharp stop, coming out to block with his upper arm, Blues then retook the lead. Andrews, who had a rumbustious afternoon, was in the right place to skewer in from close range after substitute Jack Concannon slid in onto Soldevila's cross and was blocked, the ball spinning out to the centre-forward (75th minute).
Blues then had a remarkable escape as the game swung crazily from end to end.
Albie Armin fed Brown on the right of the area and somehow Stirk managed to get in the way and head a fierce shot from zooming in the top corner.
But then it went to pot for Blues as from a Crowe cross off the right, Brown nipped it at the front post to volley past Clayton. And with two minutes left Brown waltzed through down the pitch, veered wide of Clayton and sidefooted in from a tight angle.
Blues: Clayton, Gordon (Oakley 81), Roberts, Stirk, Redmond, Campbell, Soldevila (Williams 81), Thompson-Sommers, Andrews, Fernandez (Concannon 72), George. Subs not used: Hurst, Sun.
Report: Nottingham Forest Under-23s 5-0 Blues Under-23s