Former Leeds striker Bamford turned home a rebound early in the second half at Bramall Lane for his first goal in 19 months.
Sydie Peck's penalty had given the Blades a 1-0 lead in the first half when Portsmouth defender Terry Devlin was sent off for handling on the goal-line.
Substitute Gus Hamer sealed successive wins for Chris Wilder's side, which saw them leapfrog Oxford out of the bottom three, with a brilliant curling effort from 25 yards.
The Blades secured their first win in five matches in Sunday's 3-0 Steel City derby at Sheffield Wednesday.
Femi Azeez scored on his return from injury as Millwall won 1-0 against bottom side Sheffield Wednesday at The Den.
Substitute Azeez's second-half header earned the Lions their first win in five games as they rose to fourth in the table.
Nathan Broadhead was on target as Wrexham claimed a 2-0 win over Bristol City.
Broadhead scored only his third league goal since joining from Ipswich in the summer with a stunning first-half strike.
An own goal from Robins goalkeeper Radek Vitek after George Thomason's corner struck the post wrapped up the points.
Marvin Ducksch's equaliser for Birmingham denied West Bromwich Albion in a 1-1 draw in the West Midlands derby at The Hawthorns.
Alex Mowatt's 50th career goal gave Albion a 12th-minute lead but they were pegged back by Ducksch's 78th-minute header.
Paul Smyth's late goal condemned Blackburn to a sixth home defeat of the season and earned QPR a 1-0 win at Ewood Park.