Maynard's goal, his tenth since leaving Crewe for £2.25million last summer, put Bristol City ahead but Cardiff secured a point when Ross McCormack nipped in at the foot of the post after a fumble by Basso.
It took falling behind for the Welsh side to unveil their full attacking potential and they came close to both winning and losing this Bristol Channel derby.
In stoppage time the Bluebirds owed their draw to on-loan keeper Stuart Taylor who deflected a stinging free-kick taken by Michael McIndoe from the edge of the area.
Their goalkeeping crisis saw Cardiff borrow Taylor from Aston Villa for a month and the one-time Arsenal keeper showed his skill early on with an important save in the 25th minute.
Home skipper Louis Carey fired a long low shot which was snaking inside Taylor's right post until a full dive turned it away for a corner.
Bristol's decision to play with three centre-backs had earlier allowed too much room for Paul Parry on Cardiff's left and two of his outswinging corners went close to earning them the lead.
Gabor Gyepes thundered a header over the bar from the first and Roger Johnson saw his equally assertive effort blocked on the line for the second.
After a somewhat frustrating first half in which they were too often running into offside positions, Bristol City tightened their approach with the introduction of Dele Adebola to their attack and Cole Skuse to the midfield.
They began to take control until Maynard's goal saw Cardiff step up the pace and then both Gyepes and Eddie Johnson were kept out by alert saves from Basso.