Davies took advantage of Tyrone Mings lying injured on the ground, but playing him onside, to run through and slot home an equaliser – just three minutes after David McGoldrick had scored his 12th of the season from a rebounded penalty.
Town, who made seven changes going into the game, ended strongly but did not do enough to claim the victory against a confident Preston side.
It was the League One side who started brighter, taking a run of one defeat in 15 games into the FA Cup.
Paul Gallagher was inches away from giving them a shock lead in the 12th minute but could only fire wide of the far post.
As Ipswich struggled to find any rhythm, the visitors were just a goal-line clearance away from opening the scoring midway through the half.
Jay Tabb’s woeful pass was picked up by Davies, whose cross from the right was met by strike partner Iain Hume but his effort was cleared off the line by Luke Chambers.
Ipswich were forced to live off scraps, with new signing Sylvan Ebanks-Blake going off injured just 28 minutes into his debut.
But his replacement, McGoldrick, opened the scoring seven minutes before the interval despite missing his second penalty in consecutive matches.
Paul Huntington inexplicably handled Carlos Edwards’ corner and Town’s top scorer beat Rudd at the second attempt after the keeper initially saved the spot-kick and came close to blocking the rebound.
Ipswich keeper Scott Loach then had to be alert to deny an onrushing Hume before Preston grabbed a deserved equaliser.
Davies looked yards offside when he picked up a through ball from Lee Holmes but the grounded Mings played him on and the 36-year-old calmly slotted the ball home.
The pattern of the first half, Ipswich enjoying more possession but lacking a cutting edge, continued after the break with McGoldrick unable to beat Rudd from 20 yards.
At the other end, Preston offered an occasional threat on the break and their best chance came in the 74th minute when Gallagher picked up a loose ball on the edge of the area but his rash shot was high and wide.
As the game headed towards a replay, Ipswich upped their efforts and came agonisingly close to the breakthrough in the 88th minute when Paul Taylor tricked his way to the byline but neither Frank Nouble nor McGoldrick could not divert the cross goalwards.
They came even closer seconds later as Nouble’s perfect cross was met by McGoldrick but Rudd was equal to it.