As the somewhat groggy members of the Virginia Senate staggered through dozens of bills on the next to last day of their 1978 session yesterday, they found themselves racked by indecision over compensation of dog-bite victims.

The bill, which originated in the House of Delegates, would have permitted cities and counties to pay the cost of treating dog bites suffered by their residents.

The measure was first beaten, 19 to 14, after Sen. Wiley F. Mitchell (R-Alexandria), former vice mayor of his city, complained it would plunge local governments into an injury compensation guagmire. "We don't compensate people when they get scratched by cats, run over by water buffalos or when they fall into ditches," he said. "Why should we do it for dog bites?"

When Sen. Frederick T. Gray (D-Chesterfield) posed a facetious question about the bill to Sen. John C. Buchanan (D-Wise), the only medical doctor in the Senate, Buchanan, "out of respect for turf," referred the inquiry to Sen. William A. Truban, (R-Shenandoah), the Senate's only veterinarian.

The bill finally was approved on consideration 30-3 after Sen. Elmo G. Cross, Jr. (D-Hanover) offered an amendment restricting authority for compensation to the Board of Supervisors of Spotsylvania County. He said the Spotsylvania board wanted to pay $2,000 to a woman who required extensive treatment for a dog bite.