Haynes netted either side of a Tommy Miller strike to give Ipswich a three-goal half-time lead and the tie was killed off by Alan Lee's 48th minute fourth.
Plucky Orient instantly replied through Adam Boyd and did well to restrict Town to relatively few clear chances after that.
Orient could have scored in the fourth minute when Paul Terry's free header from a John Melligan corner was cleared off the line by Alan Quinn.
Haynes and Liam Trotter threatened for Town, before Haynes opened the scoring on 20 minutes when his low angled drive beat Glenn Morris on his near post.
The keeper should have done better again two minutes later when his punch dropped to Miller 20 yards out and he coolly slotted into the empty net.
Morris did deny Jon Walters in a one-on-one, but Haynes beat him again just before half-time after racing on to Lee's pass to slot home.
Orient may have thought Town would ease off after the break and they did, but not before Lee nodded in a Quinn cross on 48 minutes.
Within seconds, Boyd scored the goal of the night with a superb 25-yard chip after seizing the loose ball in midfield.
Lee, Quinn and Trotter all sent efforts off target, while Haynes' best chance of a hat-trick came with a fierce angled drive which Morris tipped to safety.
Orient threatened occasionally through the lively Melligan down the left flank, but Town came closest to another goal when sub Kevin Lisbie got through but was denied by Morris' outstretched hand.