The hosts were handed an early lead in Saturday's Championship opener by Scott McDonald but never managed to assert their authority as Keane's men fought back with a quickfire double before Jon Stead added a third late on at The Riverside.
Three debutants in both line-ups were evidence of summers of rebuilding but it was a January arrival from north of the border who netted the opener, despite a solid first 20 minutes for the visitors.
Julio Arca - on as an 11th-minute substitute following an injury to Willo Flood - got the better of Mark Kennedy down the left before crossing for former Celtic striker McDonald, who was left in space just yards from goal and converted the simple tap-in.
The early goal spurred Gordan Strachan's men on for more, but it was the visitors who came closest to finding the net in the remaining action of the first half.
On the half-hour mark, Ipswich almost equalised as Priskin beat team-mate Carlos Edwards to head a David Norris cross against the crossbar from close range.
Aside from a weak Kennedy free-kick, that was the last action of the first half.
Immediately after the break, Ipswich were almost handed a penalty as David Wheater brought down Norris, but referee Tony Bates adjudged the contact to have taken place inches outside the 18-yard box.
From the resulting set-piece, Edwards blasted an effort at goal which took a deflection on its way out for a goal kick.
That was where the host's luck ran out, though, as just minutes later Keane's side secured the equaliser as Boro failed to clear their lines.
Smith lashed one of several clearances back at goal to eventually find the net, this time with the deflection going in Ipswich's favour.
That was on 50 minutes, and less than 60 seconds later the visitors were in the lead as under-fire Priskin opened his account for the campaign.
The 24-year-old was criticised for his contribution last season but made a better start to this campaign, latching on to an Edwards cross and heading home into the bottom corner.
Middlesbrough pushed for an equaliser but never really threatened and Stead - on as a 74th-minute substitute for Priskin - made the result safe with just over 10 minutes remaining, hitting a right-footed shot past Danny Coyne from just inside the area.