Town had lost their last three away games to stoppage-time goals, including at Burnley three weeks ago, and looked set for another let-down when Kyle Lafferty curled home a 20-yarder four minutes from time.
However, Town kept pressing forward, and when Gifton Noel-Williams handled a corner in stoppage time, Lee kept cool to send Danny Coyne the wrong way from 12 yards.
Town had had the better of the chances with Lee guilty of a horrendous miss when clean through, while Coyne also did well to deny a close-range Sylvain Legwinski header.
Ipswich started brightly when Lee got between two defenders to meet Alex Bruce's far-post cross, but his headed drifted narrowly wide.
Lewis Price in the home goal was given work to do on 11 and 13 minutes, but dealt comfortably with an angled Lafferty shot and a John McGreal header respectively.
Steve Jones sent in a dangerous low cross on 25 minutes which span into the air beyond the far post, and Jason De Vos did well to block as Wade Elliott tried to hook the ball back into the danger zone.
Chris McCann sent a 20-yarder wide before Owen Garvan, in his first home match of the season saw a fierce goalbound volley cleared by Wayne Thomas.
Jones shot against the bar, but the whistle had already gone for handball against Noel-Williams and then Ipswich enjoyed a spell of pressure.
Coyne tipped away a fierce Billy Clarke shot, saved a Garvan header and then collected an under-hit Clarke shot before Town had a penalty appeal in first-half stoppage time as Richard Naylor appeared to be blocked off, but nothing was given.
Town were quickest out of the blocks after the break and Coyne needed two attempts to hold onto a Legwinski header from a clever Gavin Williams cross right on the goal-line.
Lee should have put Town ahead on 53 minutes when Garvan and Clarke combined to send him in on goal, but he tried to bend the ball around Coyne and only succeeded in slicing way wide of goal.
Jones got in behind Richards but his attempted lob was easy for Price and, with ten minutes to go, Burnley went to one up front and seemed to have settled for the draw.
An excellent cross on the run by Richards was headed over by sub Jaime Peters from six yards and the same player then shot over after being picked out by a fine crossfield pass from Roberts.
Against the run of play, Burnley went ahead four minutes from time when Lafferty collected the ball on the right flank, cut inside Richards and curled a fine 20-yarder around Price.
A bout of pinball in the Burnley box led to a corner as stoppage time began and, after Noel-Williams was adjudged to have handled, Lee netted his ninth of the season from the spot.