The Dens Park skipper was forced out of the Clydesdale Bank Premier League clash early in the second half following what appeared to be an innocuous challenge. O'Donnell made his comeback on Boxing Day after being sidelined since August with a groin problem and could be set for another spell out of action.
Smith said: "Potentially it's a serious injury but I can't say what it is until he's been checked out. But it didn't look too good.
"I'm gutted for him because he's had a really hard time for the past four months. To come back, in his third game, and potentially have something bad happen to him is very difficult."
The defeat means bottom side Dundee remain nine points adrift of nearest rivals Ross County, whose own game fell victim to the weather. One banner at half-time at McDiarmid Park read "Smith must go," but the Dark Blues boss is determined to remain and fight for survival.
The Irn-Bru First Division runners-up were surprise additions to the top flight in the summer as late replacements for Rangers.
Smith said: "The fans are entitled to their opinion. I know what I'm doing and I know that the club will get out of the position with me there. I got stick as a player at Dundee and I came through it stronger. I'll get stick as a manager and I'll come through it stronger again.
"I've got a lot of affinity with the club. I want to make sure they stay in this league. I'll continue working as hard as ever, if not harder, to make sure we do so. We always knew it was going to be a tough, tough season and we've been proved right. We need the supporters to realise that and understand it's not going to get any easier but we will get better."
Saints boss Steve Lomas is looking forward to the winter break now after admitting the game was far from riveting.
He joked: "Some lost the will to live and I was nearly joining them. It certainly wasn't a classic. I think the accumulation of games told, I think it was about six games in 21 days. It has been a pretty hectic schedule and it's been the same 15 or 16 players who have been going week in, week out."