Ciftci, who joined the Scottish champions from Dundee United last week, missed a handful of chances to bag a debut goal but Boyata’s header and Stefan Johansen’s strike just before the hour mark ensured a comfortable win for Celtic in Wednesday’s second qualifying round, first-leg tie.
There were a couple of second-half scares for the home side in front of a partisan Celtic Park, but they will travel to Iceland next week for the second leg confident of progression, after dominating for the majority of the tie.
Ciftci’s first competitive touch for his new club nearly resulted in a debut goal less than 10 minutes in.
The forward picked up a pass from Johansen, spun his marker well 12 yards out and fired in a low shot but former Motherwell goalkeeper Gunnar Nielsen managed to block the effort with his feet.
Last year’s PFA Scotland Player of the Year Johansen then had a chance himself, but he dragged his effort wide of the post.
Stuart Armstrong was next to try his luck, but his powerful effort from 20 yards rose just over the bar.
The former Dundee United midfielder had another opportunity to break the deadlock on the half-hour mark when he latched on to Johansen’s through ball, but Nielsen was again equal to his effort.
Ciftci then got on the end of Emilio Izaguirre’s cross as the hosts threatened again but the Turkish striker could only glance his header wide.
Despite Celtic’s dominance, there were a few grumblings from the stands as the home side continued to be frustrated in front of goal.
However, the hosts did get their goal on the stroke of half-time as Boyata powered a header past Nielsen after getting on the end of Johansen’s delivery.
The Hoops could have doubled their lead before the interval when Johansen slipped the ball through for James Forrest but his effort was deflected wide.
Ciftci had another opportunity a minute after the restart but he headed Forrest’s cross over the bar.
The Scottish champions breathed a huge sigh of relief 10 minutes into the second half, however, when Jeppe Hansen fired across the face of goal for Stjarnan.
Nielsen was then called into action again for the visitors to deny Ciftci’s header after the striker was set up by Johansen.
There were more grumbles from the Parkhead crowd as the Icelandic champions created a couple of quick-fire chances in an uncomfortable period for the Hoops.
However, the fears of the home support were allayed after 56 minutes when Johansen met Armstrong’s delivery to double his side’s lead.
Celtic had a chance to all but wrap up the tie when Johansen was brought down in the box with 14 minutes remaining.
There seemed to be a bit of an argument between Johansen and substitute Leigh Griffiths as to who should take the resulting spot-kick with the latter seeming to get his way. However, his effort from 12 yards was saved by Nielsen.
There was further trouble for Celtic shortly afterwards when goalscorer Boyata had to be replaced by Efe Ambrose after the former Manchester City man went down injured off the ball.
Johansen had one final chance to seal the tie completely in injury time but he was once again denied by Nielsen in the Stjarnan goal.
Elsewhere in Champions League qualifying on Wednesday, Irish champions Dundalk scored a vital away goal in a 2-1 defeat to BATE Borisov of Belarus.
Aleksandr Karnitski broke the deadlock for the home side after 10 minutes before David McMillan equalised for the visitors just after half hour mark.
BATE will take a lead to Ireland for next week’s second leg, however, thanks to Evgeni Yablonski’s deflected effort.
The ties between Qarabag and Rudar Pljevlja, and Malmo FF and Zalgiris, both ended goalless while Dinamo Zagreb were held to a 1-1 draw at home to CS Fola Esch of Luxembourg .
Champions League second qualifying round results:
Celtic 2-0 Stjarnan
Dinamo Zagreb 1-1 Fola
Malmo 0-0 Zalgiris Vilnius
FK Qarabag 0-0 FK Rudar Pljevlja
BATE 2-1 Dundalk