Basham put his side ahead with an early close-range strike.
He then converted from the penalty spot after Scott Wiseman pulled back Ryan Harley.
Substitute McAllister hit the third after the break and in the closing stages the striker tapped home his second from close range.
Clark Keltie pulled a goal back from the penalty spot in injury time.
Against the run of play the Grecians took the lead in the 15th minute.
Steve Tully's dangerous low cross deflected off Rory McArdle and when goalkeeper Sam Russell parried the ball, Basham tapped home from close range.
Rochdale were reduced to ten men on the half-hour mark when Wiseman was judged to have pulled back Harley inside the box. Basham stepped up and made no mistake from the spot.
Adam Le Fondre should have pulled a goal back almost instantly when he was played clean through on goal but the striker was denied by anther fine save from Paul Jones.
After the break, Harley tried his luck from all of 40 yards and his shot was tipped over by Sam Russell.
Matt Gill then shot wide after swapping passes with Harley as Exeter enjoyed the extra space and controlled the game.
Exeter finished strongly and Manny Panther fired just over the top and McAllister shot just wide as the Grecians came close to extending the scoring.
The best goal of the game came when McAllister ran 20 yards and stroked the ball into the bottom corner.
McAllister struck again in the dying minutes when he tapped home a cross from Liam Sercombe.
In injury time, Dean Moxey pulled back Jon Shaw and Keltie blasted home the resulting penalty.