IRVING, Texas (AP) — Three drives, three scores. The way the Dallas Cowboys were treating the winless Cincinnati Bengals, it looked as if they’d have all afternoon to work out their kinks, especially whatever’s going on between Tony Romo and Terrell Owens.

But these Cowboys don’t do things the easy way.

They let a 17-point lead shrivel to one, then had to break up a 2-point conversion that could’ve tied it midway through the fourth quarter. Then, in the final minutes, they caught a break when a pass went through Miles Austin’s hands at the 1 and right into Patrick Crayton’s hands in the end zone, pushing Sunday’s winning margin to 31-22.

Even Dallas coach Wade Phillips said winning ‘‘doesn’t make up for everything,’’ not when the Cowboys go from losing a division game at home to nearly blowing this one.

And not when Owens catches only two passes, then delivers a postgame statement and leaves without answering any questions.

‘‘This was a great team win,’’ T.O. said. ‘‘It was frustrating out there, but I kept with it. My teammates stuck with it. … God used me today for his glory. Reality is where glory resides. That’s all I’ve got to say.’’

Owens went from having 20 balls come his way last week to having less than five this time — two catches for 67 yards, including a 57-yarder that proved to be the winning touchdown, and a run for 8 more yards. Even team owner Jerry Jones visited T.O. on the bench during the third quarter to make sure everything was OK.

‘‘I was just reminding him how important a player and an important part he is to this team winning,’’ Jones said.

Romo looked his way less partly because Dallas ran 38 times for 198 yards, with Felix Jones going for 96 and Marion Barber for 84.

But Romo also wasn’t that sharp, going 14-of-23 for a season-low 176 yards. He also had a fumble and an interception, making it eight straight games that he’s been picked off. He salvaged his day with three touchdown passes, two in the fourth quarter.

‘‘You don’t get crowned champions for being the best-looking team right now,’’ Romo said. ‘‘We know you don’t win it right now. You just continue to get better, keep stacking up wins.’’

The Bengals fell to 0-5 for the first time since 2002, when they started 0-7 and finished 2-14. Marvin Lewis took over as coach the following year and didn’t have a losing record until going 7-9 last season. Cincinnati would have to win eight of its remaining 11 games just to break even this time.

The frustrating part for the Bengals is that this was their second near-miss of a huge upset. They took the unbeaten New York Giants to overtime, also on the road.

‘‘We’re at the point where we can’t play good, we have to play great,’’ said Carson Palmer, who was 23-of-39 for 217 yards with two touchdown passes, both to T.J. Houshmandzadeh. ‘‘We have to find a way to play that perfect game.’’

Palmer missed the previous game with a sore throwing elbow and opened this game by throwing an interception. It was a deflection to linebacker Greg Ellis and it was the first of the season for the Cowboys, ending a franchise record-tying drought of five games.

Dallas only got a field goal out of it, but its next two drives ended in touchdowns — a 33-yard run by Felix Jones on a fourth-and-4, and a 4-yard pass from Romo to Jason Witten.

Down 17-0, the Bengals didn’t exactly storm back. Crawled is more like it.

They got a pair of field goals before halftime, the second set up by a Romo fumble. Then Palmer started to click, going 72 yards in 11 plays on his first drive of the second half, capping it with an 18-yard touchdown to Houshmandzadeh that made it 17-13. Then came Romo’s interception, leading to another field goal that got the Bengals within a point early in the fourth quarter.

So close to pulling off the upset, Lewis called for an onside kick. The Bengals recovered but, being the Bengals, they gave the ball back three plays later when Tank Johnson forced Chris Perry to fumble. It was Perry’s fifth in as many games.

Dallas stretched the lead when Romo hit Owens going across the middle and he ran away from everyone for the long touchdown. But the joy was short-lived because of another special-teams breakdown — Cincinnati’s Glenn Holt returning the ensuing kickoff 60 yards. Palmer followed with another touchdown pass to Houshmandzadeh. A 2-point conversion would’ve tied it at 24.

Palmer lobbed left to tight end Ben Utecht, but Keith Davis — a special teams ace starting at safety because of injuries — got enough of the ball to prevent the catch.

Crayton’s lucky grab sealed Dallas’ great escape.

Notes: Owens’ first catch was the 900th of his career. He’s the eighth receiver to do so, the third-fastest (178 games). … Chad Ocho Cinco had only three catches for 43 yards and didn’t score, so he didn’t ‘‘kiss the star’’ in the end zone as he’d hoped. … Ellis, a former defensive end, had only two interceptions in his previous 150 games.