Stephen Bradley’s side, in addition to the monetary prize of €225,000, now meet FK Mlada Boleslav of Czech Republic in the first leg of the second round at Tallaght Stadium next Thursday.
A positive Rovers start saw Burke drive over the top after incisive play on the left from skipper Ronan Finn eight minutes in. Three minutes later, Josef Jósefsson’s long throw was poorly cleared to 17-year-old Alex Hauksson whose shot took a slight deflection to bring a superb reaction save from Chencinski, turning Hilmar Halldorsson’s follow-up out for a corner.
Rovers responded superbly to that fright to take the lead with a well-worked goal on 20 minutes.
Dave Webster and Luke Byrne worked the ball from the back before Brandon Miele cleverly skipped by the challenge of Johann Laxdal on the left to cross.
The chance looked gone as the centre ran right across the face of the Stjarnan goal. Trevor Clarke thought otherwise as the winger pulled the ball back first time to give Burke a simple tap-in for his sixth goal of the season.
Minutes later, Burke found himself in space to shoot from distance to bring a full-stretch save from Haraldur Björnsson.
The Icelandic side remained a threat, Roberto Lopes doing well to deflect a shot from Eyjolfur Hedinsson.
A foul by Jósefsson on Clarke was deemed outside the box as Rovers appealed for a spot kick on 42 minutes.
Burke’s shot from the resulting free kick worked Björnsson again.
Chasing the game in need of two goals, Stjarnan enjoyed a more possession in the second half as Rovers had Chencinski to thank again on 61 minutes.
Jósefsson’s diagonal ball picked out Holmbert Fridjonsson whose cross set up Halldorsson for a first time shot the Hoops keeper somehow tipped over his bar.
With Michael O’Connor on for the hard-running Gary Shaw, keeper Björnsson tipped a Miele free kick round the post 10 minutes later as Rovers relieved the pressure.
O’Connor, twice, and Miele brought saves from Björnsson late on as Bradley’s side had done more than enough over the two legs to book their place in the next round.
Chencinski; Madden, Lopes, Webster, Byrne; Finn, Connolly (Bolger, 87); Miele, Burke (McAllister, 79), Clarke; Shaw (O’Connor, 69).
Björnsson; J. Laxdal, D. Laxdal, Gudjónsson, Jósefsson; Hauksson (Finsen, 49), Sigurdsson (Aegisson, 78), Hédinsson; Fridjonsson (Hilmarsson, 90+2), Baldvinsson, Halldorsson.
Glenn Nyberg (Sweden).