Michael Ricketts made amends for a couple of bad misses as he stroked home from 12 yards to put Burnley ahead.
But new signing Alan Lee drew Ipswich level after 62 minutes when he fired in Darren Currie's left-wing cross.
A dramatic spell saw Burnley's Wayne Thomas sent off for a second bookable offence before Michael Duff fouled Lee and Richards scored from the spot.
Burnley manager Steve Cotterill:
"In my opinion, it wasn't a penalty and that decided the game.
"Michael Duff said that Alan Lee told him it wasn't a foul as they came off
"We went ahead with a great finish from Michael Ricketts but then conceded two bad goals. If we had cleared the ball, the penalty would not have happened."
Ipswich: Supple, Wilnis, De Vos, Naylor, Barron, Westlake (Sito 45), Juan, Garvan, Richards, Lee, Currie.
Subs Not Used: Price, Magilton, Peters, McDonald.
Goals: Lee 62, Richards 83 pen.
Burnley: Jensen, Duff, Thomas, McGreal, Harley, Spicer (Elliott 89), James O'Connor, Hyde, Branch (McCann 67), Ricketts, Noel-Williams (Courtney 83).
Subs Not Used: Karbassiyoon, Garreth O'Connor.
Sent Off: Thomas (81).
Booked: Thomas, Ricketts, Spicer, Elliott. Goals: Ricketts 45.
Ref: D Deadman (Cambridgeshire).