The win was set up in the 27th minute when the Scotland striker curled in a textbook free-kick from 25 yards to reach his century.
Midfielder Callum McGregor added a second seven minutes after the break before defender Kristoffer Ajer headed in his first goal for Celtic just after the hour mark to complete a comprehensive 4-1 aggregate win.
After being knocked out of the Champions League qualifiers by AEK Athens, the Northern Irishman had stressed the need for European football until at least Christmas and his side delivered that with some style, albeit against limited opponents.
The Scottish champions join Rangers in Friday's draw and both Glasgow clubs being in Europe will add some more spice, if needed, to Steven Gerrard's first Old Firm game as Ibrox boss on Sunday.
Whether Moussa Dembele features is another question.
Before the game Rodgers revealed Celtic had rejected a bid for the French striker, with Lyon having confirmed their interest in the 22-year-old on Wednesday night, two days before the deadline.
Griffiths led the line with Scott Sinclair and Mikey Johnston, while Olivier Ntcham returned to the starting line-up.
It was soon evident that Suduva were happy to defend their box in numbers but a lapse when trying to deal with a Sinclair cross in the 18th minute allowed Griffiths possession but his shot on the slide from 12 yards was straight at goalkeeper Ivan Kardum.
Griffiths' milestone goal came after Jovan Cadenovic fouled McGregor 25 yards from goal after the Hoops midfielder had worked hard to regain possession.
The former Hibs forward stepped up and curled a wonderful left-footed shot past the flailing Kardum to become the first Parkhead player to reach the 100-mark since John Hartson in November, 2005.
Celtic should have doubled their lead in the 33rd minute when Ntcham's cross from the right ended up at the unmarked Sinclair 10 yards from goal but Kardum was quick off his line to block the shot.
McGregor's well-take goal in the 52nd minute effectively sealed Celtic's passage into the competition proper.
Sinclair drove in from the left and laid the ball off to McGregor, who took a touch before drilling a shot from the edge of the box low past Kardum.
The home side piled forward in the search for more goals and Griffiths missed the target from close range before being denied his second by a fine tackle from Suduva defender Algis Jankauskas.
The third goal arrived when Ajer dropped to the back post to head a Griffiths delivery back across and in off the far post.
Suduva became increasingly bedraggled as Celtic missed a series of chances before the end.
Notably, Johnston drove straight at Kardum on a one-on-one before his replacement Jonny Hayes had the ball in the net only to be ruled offside.