The victory means the Gunners move to within a point of Manchester United, whom Arsenal have played two games fewer than, in fourth and only sit behind fifth-placed West Ham - who have played three games more and could only draw with Newcastle earlier in the day - on goal difference.
Forward Alexandre Lacazette saw an effort ruled out for offside in the first quarter of an hour, while his penalty claims in the build-up were also waved away.
Further shouts for a spot-kick were again dismissed five minutes before the break when Cedric Soares's shot struck the arm of the sliding Yoane Wisa.
Arsenal did not have to wait long after the restart to open the scoring as Smith-Rowe showed quick feet before firing in on 48 minutes before Saka sealed the victory 11 minutes from time with a sweeping counterattack.
Brentford, who felt defender Pontus Jansson was fouled in the build-up to the second, pulled one back through Christian Norgaard in stoppage time, however, it was not enough to stop Thomas Frank's side from slipping to their sixth defeat in seven games.