Good work from Christian Nade allowed Stewart an easy finish for his seventh goal of the season late in the first half.
Motherwell fell to their fifth Scottish Premier League defeat in a row but they put Hearts under fierce pressure for much of the second half.
Hearts withstood waves of attacks to record their third consecutive win as the visiting players responded well following the sacking of Jim Gannon.
Former Scotland manager Brown, 69, began work on the eve of the game after the Fir Park board lost patience with Gannon's unwillingness to sign a contract six months into his reign.
The short turnaround meant youth development chief Gordon Young picked the team and he made three changes from Saturday's 3-1 defeat by St Johnstone.
Stevie Hammell, Ross Forbes and Jamie Murphy came in with Yassin Moutaouakil missing out through injury and Keith Lasley and John Sutton dropping to the bench.
Brown, who began his coaching career at Fir Park in 1974, took a place on the bench alongside assistant Archie Knox and they should have been celebrating inside two minutes.
John Ruddy's long kick caused confusion in the home defence and Marius Zaliukas headed the ball straight to Jim O'Brien with Marian Kello way off his line, but the winger's cushioned volley lacked power and allowed the goalkeeper to get back just in time to claw the ball off the line.
Murphy and O'Brien, playing in advanced wide positions in Motherwell's five-man midfield, both had decent efforts at goal as the visitors continued their bright start.
Well threatened with a direct counter-attack with Lukas Jutkiewicz inches from connecting with O'Brien's left-wing cross.
Hammell comfortably headed out a Jose Goncalves header as he guarded the near post at a corner before Hearts lost Jamie Mole to injury in the 20th minute, with Nade coming on.
Hearts pushed Motherwell back for a 10-minute spell but a wayward shot from Nade was all they had to show for their pressure.
Motherwell looked comfortable but they conceded a soft goal in the 39th minute. Nade turned Stephen Craigan on the byline and recovered quickly after the Well captain and Tom Hateley combined to stop him crossing.
Ruddy came well out of his goal to close Nade down but the goalkeeper was stranded when the Frenchman cut the ball back and Stewart passed it into the net from eight yards.
Forbes volleyed well over from the edge of the box after Jutkiewicz had intercepted a poor pass back from Zaliukas moments after the restart but Motherwell almost fell further behind in controversial circumstances in the 54th minute.
Crawford Allan gave Motherwell a free-kick on the halfway line after Steve Jennings and Stewart went in for a challenge but reversed his decision and allowed Goncalves to take it quickly with the visitors up field.
Nade was clear down the left and crossed for Suso Santana but the Spaniard skewed his volley high and wide before the Motherwell players surrounded the referee and Brown raced to the touchline to protest.
Motherwell pushed forward and Kello kept out a Steven Saunders header before diving low to brilliantly push away Jutkiewicz's strike following a knockdown by Sutton, on moments earlier form Murphy.
The Hearts keeper then held a Sutton volley and smothered at the feet of the Englishman after Jutkiewicz and O'Brien had worked their way into the box.
Motherwell brought on Lasley and Steven McGarry, making his first SPL appearance of the season, and they continued to press.
But Hearts held firm with their defence putting their bodies in the way time after time, especially during a late goalmouth scramble when Kello came off his line to block from Lasley.
Craigan then headed wide from a Forbes corner and Kello beat away a Forbes free-kick deep into injury time.