Curtis stood at two-under-par 208, one stroke better than fellow American J.B. Holmes, the overnight leader, and Sweden's Henrik Stenson.
Because of the downpours and lightning that hit the Detroit suburb on Saturday and suspended third-round play, the leaders were having to navigate 36 holes on Sunday.
The fourth round of the year's final major championship was scheduled to begin at 1220 local time (1620 GMT) with Curtis, Stenson and Holmes teeing off in the final group at 1410 (1810).
Players will compete in threesomes and go off from the first and 10th tees in order to fit in the final round.
Stenson birdied the 18th to reach one-under-par 209 with a round of 68, while Holmes bogeyed the last to complete an even-par 70.
Curtis, the 2003 British Open winner, also bogeyed 18 after hitting his tee shot into a bunker but had built up a strong round on the rain-softened course with four birdies over his first six holes during the chilly morning.
Curtis, Stenson and Holmes were among six players who did not hit a tee shot on Saturday, when play was called off after a delay of more than four hours and only 25 of the 73 players left in the field were able to finish the third round.
Double British Open champion Padraig Harrington surged up the leaderboard by posting a four-under-par 66 to stand three shots off the pace at one-over 211 along with Spain's Sergio Garcia and Charlie Wi of South Korea.
Garcia shot 69, and the 36-year-old Wi, playing his first major, returned a 71.
Leading third-round scores:
208 — Ben Curtis (US) 73, 67, 68.
209 — Henrik Stenson (Sweden) 71, 70, 68; J.B. Holmes (US) 71, 68, 70.
211 — Padraig Harrington (Ireland) 71, 74, 66; Sergio Garcia (Spain) 69, 73, 69; Charlie Wi (South Korea) 70, 70, 71.
212 — Andres Romero (Argentina) 69, 78, 65; Jeev Milkha Singh (India) 68, 74, 70.
213 — Camilo Villegas (Colombia) 74, 72, 67; Steve Flesch (US) 73, 70, 70; Aaron Baddeley (Australia) 71, 71, 71; David Toms (US) 72, 69, 72.
214 — Prayad Marksaeng (Thailand) 76, 70, 68; Graeme McDowell (Britain) 74, 72, 68; Phil Mickelson (US) 70, 73, 71; Angel Cabrera (Argentina) 70, 72, 72; Justin Rose (Britain) 73, 67, 74.
215 — Retief Goosen (South Africa) 72, 74, 69; Stuart Appleby (Australia) 76, 70, 69; Alastair Forsyth (US) 73, 72, 70; Ken Duke (US) 69, 73, 73.
216 — Ernie Els 71, 75, 70; Fredrik Jacobson (Sweden) 75, 71, 70;
Robert Karlsson (Sweden) 68, 77, 71; Michael Allen (US) 70, 75, 71; Brian Gay (US) 70, 74, 72; Briny Baird (US) 71, 72, 73; Paul Goydos (US) 74, 69, 73; Nicholas Thompson (US) 71, 72, 73; Brandt Snedeker (US) 71, 71, 74.
217 — Justin Leonard (US) 74, 71, 72; Steve Elkington (Australia) 71, 73, 73.
218 — Jim Furyk (US) 71, 77, 70; John Merrick (US) 73, 75, 70; Paul Casey (Britain) 72, 74, 72; Ian Poulter (Britain) 74, 71, 73; Rory Sabbatini (South Africa) 72, 73, 73; Sean O'Hair (US) 69, 73, 76.
219 — Mike Weir (Canada) 73, 75, 71; Chris DiMarco (US) 75, 72, 72; Rocco Mediate (US) 73, 74, 72; Anthony Kim (US) 70, 75, 74; Peter Hanson (Sweden) 71, 73, 75; Michael Campbell (New Zealand) 73, 71, 75; Tom Lehman (US) 74, 70, 75; D.J. Trahan (US) 72, 71, 76.
220 — Robert Allenby (Australia) 76, 72, 72; John Senden (Australia) 76, 72, 72; Charl Schwartzel (South Africa) 77, 70, 73; Mark Brown (US) 77, 69, 74.—Reuters