Just eight minutes in, Olivier Giroud nearly scored a spectacular goal as he attempted a scissor kick, but his well-executed left-footed effort cannoned off the bar.
The Frenchman then came close again as he planted a header onto the roof of the net, and in the 25th minute, the Gunners went on to finally break the deadlock.
Alexis Sanchez sprayed a pinpoint crossfield pass into the Doncaster box for Walcott, who brought the ball down and rifled into the net.
Just one minute after the restart, Reiss Nelson cut inside from the right and had a go, but his rising effort sailed well over the bar.
Arsenal then came close again in the 50th minute as Giroud threaded the ball into the path of Walcott, but the winger's effort when clean through on goal clipped Ian Lawlor and went over the bar.
However, the Gunners managed to hang on until the final whistle, which secured their passage to the fourth round of the competition.