Eddie Howe's Clarets had taken one point from their past two games, but Rodriguez's 18th goal of the season in the third minute steadied the nerves at Turf Moor.
Barnsley only threatened sporadically and boss Keith Hill had used up all three of his substitutions as the second half got under way.
They nearly fashioned a way back into the the contest when Craig Davies rattled the home crossbar.
However, Burnley responded and gave themselves breathing space, with Charlie Austin on hand to end a six-game personal goal drought when Rodgriguez's 63rd minute cross came back off the post.
Howe's side have edged to within four points of the top six, while back-to-back defeats mean Barnsley remain lodged in lower midtable.
Scott Golbourne came into the Barnsley defence for his full debut, while Millwall loanee Josh McQuoid replaced Keith Treacy in the only change to the Burnley line-up.
And McQuoid was heavily involved as the host made a flying start.
Ross Wallace cut back inside onto his left foot to cross from the right wing, and McQuoid's knockdown at the far post left Rodriguez with a simple finish for his fourth goal in as many matches.
Wallace was causing Barnsley plenty of early problems down the right flank where he was ably supported by full-back Kieran Trippier, who stung the palms of visiting goalkeeper Luke Steele with an angled drive in the 12th minute.
Shortly afterwards Burnley's fans rose to give a warm ovation to club chairman Barry Kilby. The 62-year-old announced last week that he will step down at the end of the season after being diagnosed with cancer.
With 17 minutes gone, a crisp one-touch move involving Wallace, Rodriguez and Trippier ended with a backpedalling Austin heading over the Barnsley bar.
Shortly before the half hour Burnley captain Chris McCann appealed for a penalty after a coming together with Tykes defender Scott Wiseman on the edge of the area, but referee Oliver Langford was unimpressed and responded by booking the Northern Ireland international for simulation.
The officials further irritated the home faithful in the 43rd minute when David Perkins appeared to clear Austin's header from Trippier's corner from behind the line, but the goal was not given.
Howe's side did not allow the controversy to knock them out of their stride and after the resumption two delicious crosses from Trippier were headed narrowly off target by McCann and Rodriguez.
But Barnsley were threatening to edge their way into the match and Davies almost gave them an equaliser in the 56th minute when he thundered a 25-yard strike against the bar.
Burnley heeded the warning and the woodwork also played a part as they doubled their lead.
Dean Marney stormed forward from midfield to released Rodriguez and, when he hit the post, Austin was on hand to apply the finish.
McQuoid fizzed a volley just past the post before Burnley played out the closing stages comfortably and handed a debut to 19-year-old striker Danny Ings, who came on as substitute to end a six-month knee injury nightmare following his move from Bournemouth in August.