Although Messi had been considered a doubt after sustaining a thigh injury against Villarreal, he came straight into the line-up for his second start of the season and set Suarez up for his 84th-minute decider, as Barca just about did enough to see off the Serie A leader.
An Argentinian No.10 was on the scoresheet in only the second minute of the match, but it was not the returning Messi, as Lautaro Martinez got his first ever Champions League goal.
Barca – dominated in the first half – took charge after the interval and restored parity through Suarez's fine volley.
The Uruguay striker completed the turnaround six minutes from time with a similarly gorgeous effort after Messi's wonderful run.
Inter enjoyed a lightning start that was a sign of things to come in the first half and took the lead in the second minute – Martinez racing clear and holding off Clement Lenglet before finding the bottom-right corner.
Nelson Semedo made a vital block to deny Alexis Sanchez just past the half-hour mark, a few minutes before Marc-Andre ter Stegen produced a stunning save to keep Martinez's goal-bound header out.
Barca showed signs of improvement after the break and got the leveller just before the hour, Suarez lashing a terrific volley down to Samir Handanovic's left.
Griezmann should have at least tested Handanovic soon after but sliced into the side-netting after being played into the left side of the box by Messi.
However, Suarez ensured Messi's excellent performance was not for nothing, as Barca's talisman found the Uruguayan on the edge of the box and he slotted home the winner in nonchalant fashion after an exquisite touch left his compatriot Diego Godin for dead.