The on-loan Sunderland striker stabbed Boro ahead just before the half-hour mark - his team's first goal in eight games, totalling 12 hours 14 minutes - and then headed home a second just before the interval.
Boro had a good early chance as Emmanuel Ledesma played a one-two with Graham on the edge of the box before firing a low shot just wide.
Then, on 12 minutes, when a Ledesma free-kick was nodded down by Ben Gibson, George Friend fired over the bar.
Soon after, Mustapha Carayol sent Friend overlapping down the left to cut inside and get into the box but once again he scooped high over the bar.
Ipswich went close on 28 minutes as a Daryl Murphy shot hit defender Kenneth Omeruo and fell at the feet of Paul Taylor 12 yards out but he also lashed over.
Boro finally ended their long wait for a goal in the 29th minute.
Carayol cut in from the left to fire a stinging shot in and, when goalkeeper Dean Gerken spilled it, the ball feel nicely for Graham to ram home from close range.
And the Teessiders got a second in first-half stoppage time as former Ipswich man Grant Leadbitter whipped in a corner and Graham glanced in a header.
After a slow start to the second period, the visitors threatened just before the hour, Paul Taylor sending an effort bouncing wide through a crowd of players following a free-kick.
And soon after, Paul Green carried the ball forward unchallenged only to put a speculative shot from distance well wide.
Ipswich had a clear sight of goal on 66 minutes as Green crossed to the edge of the box for Murphy to turn quickly, but his shot was weak and straight at Boro goalkeeper Dimitrios Konstantopoulos.
Boro stepped up a gear and went close on 76 minutes as Lee Tomlin sent a 20-yard effort just over.
Then on 79 minutes, Kei Kamara streaked down the right and cut inside to drill in a 20-yard attempt which came back off the shins of a defender.
The home team almost snatched a third in stoppage time as Kamara broke down the right and crossed to the far post, where substitute Curtis Main had arrived unmarked, but he could only find the side-netting from close range.