This tie had been due to take place at Wimbledon's Plough Lane home last week before flooding caused damage to the pitch.
It was hastily rearranged at St James' Park instead, with Saudi-backed Newcastle donating £15,000 to Wimbledon to aid the substantial repair effort.
'Thank you NUFC and the wider football family' said a banner held up in the away end by Wimbledon fans.
Newcastle's charity did not stretch to matters on the pitch, though, as they clinically closed out the tie against a side 60 places below them in the football pyramid.
Howe made eight changes from the team which held Manchester City to a 1-1 draw at the weekend but Wimbledon were comfortably second best to the Magpies' second best.
The Dons were organised, disciplined and showed plenty of fighting spirit but carried next to no threat in the final third.
As soon as Newcastle got their noses in front there was no way back into the contest for Johnnie Jackson's side.