The full-back swept home Eden Hazard's cross to make it two wins from two under Maurizio Sarri and keep the Gunners waiting for their first Premier League point under Unai Emery.
It was a particularly galling end to the game for the visitors, who had dragged themselves level before the break thanks to Henrikh Mkhitaryan and Alex Iwobi, after Pedro and Alvaro Morata had put Chelsea two goals to the good.
Manchester City went ahead against the Gunners after 14 minutes last week, and Chelsea needed just nine to find the net, with Pedro finishing coolly from 12 yards after being teed up by Alonso.
It was 2-0 after 20 minutes when Morata turned away from Shkodran Mustafi and fired low past Petr Cech, just a minute after Pierre-Emerick Aubameyang, starting as the lone striker once more, had blazed over the crossbar with the goal gaping.
Mkhitaryan produced almost a carbon copy of Aubameyang's miss but made amends with a fine left-footed strike to get the away side back into the game, and Iwobi thumped high into the net from the Armenia winger's cross to level the scores before the break and leave the home crowd shell-shocked.
The Blues suddenly looked in disarray and could have been behind heading into half-time had Aubameyang and Iwobi been more accurate when finding space among a ragged Chelsea defence.
It was Sarri's men who looked the most likely to break the deadlock in a more scrappy second half, though, and they did just that with 81 minutes played, as Alonso finished with aplomb following fine work from substitute Hazard.
