An early goal from Steph Houghton put the reigning Super League champions on their way to victory and further goals from Jordan Nobbs and Ellen White sealed a convincing win
Steph Houghton ensured the Gunners got off to the perfect start by heading home Rachel Yankey's cross in the second minute.
And Jordan Nobbs added a second midway through the second half to seal victory in a match the north London club controlled before Ellen White tapped in a late third.
Arsenal were missing experienced England international Kelly Smith due to injury, so White started alone in attack, while Bristol top scorer Laura del Rio was fit enough only to take a place on the bench.
Shelley Kerr's team have stuttered in their opening two matches in the Women's Super League this season, collecting only one point from a possible four, but got off to the perfect start in South Yorkshire.
Yankey was allowed too much space on the left-hand side and floated a cross to the back post which Houghton, who had burst forward from central midfield, carefully guided past Siobhan Chamberlain.
Bristol were nearly gifted an immediate reply when Kim Little, believing a corner from the right was heading out of play, allowed the ball to bounce off the near post but Jemma Rose could not keep her shot down.
From then on, Arsenal were in control. Ellen White made contact with another superb Yankey delivery to force Chamberlain into a tipped save before Nobbs hit the post after a surging run.
Chamberlain was called into action again before half-time by Gunners right-back Alex Scott, who saw her powerful 25-yard drive turned around the far post.
Bristol held on until half-time, though, and had made an encouraging start to the second half before they were dealt a double blow.
After Lucy Staniforth had been forced off in some pain with a knee injury, the impressive Nobbs doubled Arsenal's lead with a looping header after Gemma Davison had crossed to the back post.
Bristol threw bodies forward in the closing stages in search of a goal that would reignite the game but Arsenal continued to have the better of the chances as gaps opened up on the counterattack.
White had been denied all afternoon by Chamberlain, without whom Arsenal's lead could have been much more comfortable, but finally grabbed a deserved third by pouncing on a rebound after the goalkeeper had superbly denied Nobbs from inside the box.