Burns contributed 24 points including a 61st-minute try as the West Country outfit came from 13 points behind early in the second half.
Sailosi Tagicakibau crossed for Irish in the second minute then Halani Aulika and Anthony Watson scored tries within three minutes of one another after the break.
However replacement prop Dan Murphy ran in a try for the visitors before fly-half Burns came into his own.
The two conversions and a fifth penalty earned the win.
Bayonne were held 13-13 at home by Wasps as the Premiership side retained top spot in Pool 3. Also in that group, Newport Gwent Dragons had no trouble in running out 33-0 winners over bottom Italian side Mogliano to stay in contention.
Bath continued their 100 per cent record in Pool 4 with a 67-11 win over another Italian outfit in Calvisano while Romania's representative Bucharest Wolves shocked Agen 25-22 to go second ahead of the third-placed Top 14 club.
Another French side Stade Français won a third game from three in Pool 5 against bottom side Cavalieri Prato, but it was not plain sailing against the Italians as the match finished 37-23.
Spanish side Gernika won the battle of the Pool 2 strugglers 13-3 over Rovigo of Italy.