England resumed on 195 for four, a lead of 11 runs, after Kevin Pietersen had shown something of his old form with a fluent 85 on Saturday.
Pietersen had come in with England struggling at 14 for two, following the early loss of captain Andrew Strauss and Jonathan Trott.
But Pietersen, together with opener Alastair Cook (55) put on 106 for the third wicket and then shared a stand of 71 with Bell for the fourth.
Bell was 39 not out overnight and Anderson unbeaten on nought.
In a match where 155 overs had already been lost to rain, England were keen for quick runs in a bid to force a victory that would give them a 2-0 series win after an innings and 14-run victory in the first Test in Cardiff was followed by the drawn second Test at Lord's.
Anderson drove the third ball of the day, from Thisara Perera, off the back foot through the covers for four.
Next ball Anderson, who bats left-handed but bowls right, gently steered Perera through the slip cordon for another four.
Later he followed up with another superb back-foot cover drive for four off Perera -- a shot worthy of any 'proper' batsman.
Bell then completed a 74-ball fifty with his eighth boundary, a deftly-timed late cut off seamer Suranga Lakmal.
Anderson struck two successive cover-driven boundaries off Welegedara but, next ball, he was caught behind off the left-arm paceman, who has repeatedly troubled England captain and opening batsman Andrew Strauss this series.
England were now 236 for five, a lead of 52 runs, but a Sri Lanka attack that has struggled throughout this series rarely looked like taking another wicket in the session.
