The Spain midfielder, who missed three matches with a hamstring injury, scored his 10th league goal of the season in the 28th minute after a neat one-two with Eduardo.
His surging run then led to substitute Fran Merida scoring the second 12 minutes from time as former Wanderers player Coyle's reign got off to a losing start.
Arsenal's victory, which saw then soak up plenty of pressure from a struggling Bolton side trying to play their way out of trouble, lifted them to within three points of league leaders Chelsea.
With these two teams due to meet in the return fixture at the Emirates Stadium on Wednesday, Arsene Wenger's side could well take over at the top in the next few days.
The Gunners, unbeaten in the league since the end of November, owed everything to their inspirational captain, who, even though his influence faded in the second half, was instrumental in the win.
How Coyle, who swapped one relegation battle with Burnley for another with Bolton, must wish he had a player with half the drive and invention.
The paucity of resources available to the Bolton boss was evident by the fact he named exactly the same starting XI and bench as predecessor Gary Megson had done in his final game - the 2-2 league draw with Hull on December 29.
Initially he will have been pleased by what he saw as the new manager factor had an immediate effect with Kevin Davies having a close-range shot blocked and Ivan Klasnic drilling a 20-yard effort past Manuel Almunia only to be flagged offside in the opening two minutes.
The transition from Megson's long-ball tactics to Coyle's football is something that will obviously take a while to get used to but there were fleeting glimpses of the 'new' Bolton.
Davies, Gretar Steinsson and Klasnic combined down the right in a short passing move to set up Tamir Cohen 20 yards out but he fired well over.
Arsenal, with Fabregas at the heart of everything, had struggled to create a clear opportunity until they took the lead in the 28th minute.
Steinsson slipped on the edge of the penalty area to hand possession to Fabregas and the Spain international capitalised, quickly exchanging passes with Eduardo before slotting a low shot past Jussi Jaaskelainen's outstretched right hand.
Arsenal's 19-year-old midfielder Craig Eastmond, making his first league start, almost doubled the lead four minutes later with a first-time shot from Fabregas' corner, which Jaaskelainen saved on the line with his legs.
And the excellent Gunners captain should have had a penalty in the 33rd minute when he was caught by Zat Knight but referee Phil Dowd waved away claims.
Two minutes before the interval Fabregas missed his easiest chance from only seven yards from Andrey Arshavin's cut-back before Klasnic mis-kicked from even closer range in added time at the end of the first half.
Bolton began the second half as they had the first, with both Matt Taylor and Cohen causing Almunia concern in the early stages.
Lee Chung-Yong, Wanderers' best player, had caused Armand Traore problems all game and, having out-stripped the left-back once again, his low cross was deflected goalwards by William Gallas but Almunia collected.
Taylor was guilty of not testing the Arsenal goalkeeper in the 61st minute when Klasnic spun on the edge of the area to release him on the left only for his team-mate to blaze over.
Taylor was again the culprit in the 70th minute when he sprang the offside trap from substitute Gavin McCann's lofted pass but tried to beat Almunia from 35 yards despite being well clear of the nearest defender and ended up shooting wide.
Arsenal's response was for Thomas Vermaelen to curl a 25-yard free-kick on to the top of Jaaskelainen's net.
Fabregas had been quieter after the break but, when he did take control again, it set up the second goal 12 minutes from time.
His surging run into the penalty area allowed Eduardo to cross to the far post where Merida side-footed home to clinch victory.