Bendtner should have equalised for Arsenal after 53 minutes when he arrived on the endof Gael Clichy's cross, but his header from eight yards was brilliantly turned away by Valdes.
Just after the half hour mark United had a half chance when defender Darren Patterson found himself on the endof a David Hannah overhead kick, but his header was easily saved by Saints' keeper Alan Ferguson.
Boyd sent a mis-cued header over the bar from eight yards out towards the endof the first half, with the visitors leading though Sebastian Rudy's close-range strike.
Blues midfielder Gardner ran unmarked on to the endof a whipped-in Stephen Carr cross and directed a firm downward header which Reina somehow got a hand to.