Argyle took the lead when Antoni Sarcevic swept in from the right and let fly with a rising drive that took a deflection to beat City goalkeeper Colin Doyle.
Earlier Doyle had made a fine save to deny in-form Argyle attacker Ruben Lameiras.
Bradford responded well with Paul Taylor twice forcing saves from Plymouth stopper Remi Matthews.
The first half ended with Argyle in the ascendancy, boosted by Sarcevic’s goal and a stinging Lameiras low drive that flew inches wide.
Argyle playmaker Graham Carey came close to doubling the home side’s advantage at the start of the second half but shot over after beating three players in the box.
A brilliant Bradford goal-line block by skipper Romain Vincelot denied Sarcevic as he let fly from close range after latching on to Sonny Bradley’s nod down from a Carey corner.
Bantams striker Charlie Wyke was sent off following a straight red card in the last minute of normal time after an aerial challenge with Bradley.
England manager Gareth Southgate has claimed that the country needs to boot racism out of its own football before criticising World Cup hosts Russia.
Newcastle have been linked with a shock move for former Sunderland goalkeeper Simon Mignolet during the summer.