The 25-year-old, who arrived from Middlesbrough last month, continued to press his claim for a starting spot in Burnley's opening Premier League fixture against Chelsea on August 18, taking his tally to five in as many matches.
A Burnley side missing key man Danny Ings took the lead through Jutkiewicz in the second minute.
After Alex Lopez had equalised for Celta midway through the second half, Jutkiewicz responded immediately with a back-post tap-in.
However, the visitors were the better side for much of the second period and got the draw their play merited when Nolito finished off a fine team move 17 minutes from time.
Burnley made a flying start as Jutkiewicz received a pass from Ross Wallace and showed his strength to hold off the challenge of a defender before slotting the ball beyond Ruben Blanco.
Jordan Dominguez headed Samuel Araujo's left-wing cross straight at Tom Heaton in the 13th minute as Celta recovered from the early setback with a period of sustained possession.
Yet Jutkiewicz twice came close as the hosts continued to look the more dangerous, striking the post with a scuffed effort and seeing another shot kept out by the legs of Blanco.
Those two chances sandwiched a near miss for Ben Mee, who sent a powerful header narrowly wide of the post from Wallace's outswinging corner.
Tempers threatened to boil over shortly after the half-hour with Celta's Santi Mina shown a yellow card following an altercation with Dean Marney, which saw several players rush over in an attempt to keep the peace.
Celta cut Burnley open for the first time in the 37th minute, but Heaton spread himself to deny Joaquin Larrivey before Kieran Trippier blocked Dominguez's follow-up.
The visitors made several changes at the break and were almost level immediately after the restart when Fabian Orellana struck the crossbar with a fine curling effort from the edge of the area.
Looking much improved in the second half, Celta were dominating proceedings and saw Charles head a pinpoint Orellana cross over the bar from six yards in the 64th minute.
Heaton then made a smart save from Nolito as Burnley's lead looked increasingly vulnerable.
The equaliser duly arrived in the 69th minute, Lopez taking a touch and planting a finish beyond Heaton from 10 yards.
Parity lasted only a minute, as Jutkiewicz was left with the simple task of tapping Ashley Barnes' driven cross into an empty net, but Nolito had the final say as the Spanish outfit departed Lancashire with a draw.