Clarke, coming off his two lowest scoring outputs of the season, led all players with 28 points, including 15 in the second half. His fifth three-point field goal of the game with 7:42 remaining put the Bulldogs in front for good, 57-56.
Smith, who had not scored in double-figures in Butler's previous three games, posted his second double-double of the season with 24 points and 10 rebounds. The 6-11 center scored 14 points after intermission when the Bulldogs out-scored the host team, 37-26.
"Certainly, these two guys carried us tonight," Butler coach Brad Stevens said in a release. "Andrew really settled us down. And Rotnei, every time they'd cut it to two it seemed like Rotnei would hit one and get it back to five or hit one to tie it up. These guys just made plays that seniors make."
The Hawks (8-5, 0-1 A-10), who were tabbed as the preseason favorite in the A-10, looked primed to pick up their fourth consecutive win in the early going. Behind the long-range shooting of sophomore guard Chris Wilson, Saint Joseph's jumped to a 12-5 lead and then stretched the margin to as many as eight points, 35-27, late in the opening period.
Wilson, who had just seven three-point field goals in the Hawks' first 12 games, nailed five of seven shots from beyond the three-point arc in the first 20 minutes. As a team, Saint Joseph's hit eight of 13 three-pointers in the first half, while building a 40-35 halftime lead.
It was Butler that came out strongly in the second half. The Bulldogs opened the final period with a 9-0 run behind two Smith baskets, a three-point field goal by Clarke and a steal and basket by sophomore Alex Barlow.
Saint Joseph's did climb back on top by five points on two occasions, the last coming at 53-48 on a pair of free throws by forward Halil Kanacevic, but the Bulldogs answered with a 13-3 run that carried Butler to a 61-56 advantage. Clarke scored seven of the 13 points for the Bulldogs in the decisive run.
"I was really proud of our guys for staying the course," Stevens said. "I thought (we) executed a plan absolutely to a 'T', and it went a little bit awry in the first half with Wilson and (C.J.) Aiken in the second making all those shots. We were going to guard (Carl) Jones and (Langston) Galloway as hard as we could, and you have to give up something against really good teams. We were going to be a little late to Wilson and Aiken and the other 'bigs', and boy it almost burned us."
Butler hit 13 of 27 (.481) shots in the second half, including three of six from beyond the three-point arc. And the Bulldogs limited Saint Joseph's to 25.9 percent (7-27) shooting, including just two of 12 from long-range, after intermission. The Hawks had one nine-minute stretch without a field goal in the second half.
Clarke, who had just 10 total points in Butler's previous two games, posted his highest scoring total of the season. He also matched his season rebounding-high with six and added four assists.
Smith, who hit eight of 15 shots from the field and was perfect on seven free throw attempts in the game, led both teams in rebounding. The Bulldogs finished with a 40-37 rebounding edge for the game.
Wilson led four Saint Joseph's players in double-figures with 15 points. Aiken added 13, Ronald Roberts had 12 and Galloway finished with 11. Jones, the Hawks' leading scorer, finished with seven.
Butler's will continue its opening A-10 road trip with a game at Dayton on Saturday at 2 p.m.