The Pirates (8-4, 7-1) will advance to play Tulsa for the league championship if Alabama-Birmingham beats Central Florida on Saturday.
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Carden scored after three straight runs by Vintavious Cooper took the ball down to the Marshall 1.
East Carolina had stopped a Marshall possession earlier in the overtime when Chrishon Rose forced a fumble at the 11 by Essray Taliaferro and Derrell Johnson recovered it.
A 25-yard touchdown pass from Carden to Andrew Bodenheimer gave East Carolina a lead on the first play of overtime, but the Thundering Herd (5-7, 4-4) answered with a 4-yard Blake Frohnapfel scoring pass to Antavious Wilson after Taliaferro converted a fourth-and-1 with a 4-yard run.
Frohnapfel took over at quarterback for Marshall after Rakeem Cato suffered an injury to his lower left leg midway through the third quarter. He completed 12 of 15 passes for 101 yards and raced 51 yards for a fourth-quarter touchdown, the longest run allowed by East Carolina this season.
Carden drove the Pirates 76 yards in 11 plays to tie the game with 4 seconds left in regulation on a 6-yard pass to Danny Webster. The drive, which started with 1:55 left, included three completions to Justin Hardy, who caught a school-record 16 passes for 171 yards, fourth-best in East Carolina history. His 1,046 receiving yards for the season is also fourth-best in school history.
The 124 combined points were the most ever in a Conference USA game, although the two schools combined for 125 in the 2001 GMAC Bowl, when Marshall played in the Mid-America Conference. The Thundering Herd won that one 64-61 in two overtimes.
East Carolina's 65 points were its fourth-highest total ever and the most it has scored since a 66-23 victory over East Tennessee State in 1981.
Marshall's 59 points were the most it has ever scored in a loss.
Carden completed 38 of 47 passes for 439 yards and three touchdowns and ran for three more, including the deciding score in overtime. His first two touchdown runs helped stake the Pirates to a 28-7 second-quarter lead.
The Thundering Herd rallied to trail 35-28 at halftime after three second-quarter touchdown passes by Cato. He threw five in the game, tying his career high, and his 318 passing yards before being injured gave the sophomore 4,201 for the season. It was the fourth time a Marshall quarterback has surpassed 4,000 yards.
Tommy Shuler caught 14 passes for 141 yards and two touchdowns for Marshall, giving the sophomore a school-record 110 for the season.
Taliaferro, in his first start this season, rushed for 130 yards on 27 carries for the Thundering Herd.